Odd-Shaped Panel

A Comic Book/Comic Strip/Manga/Web Comic panel takes on an irregular shape as a form of Painting the Medium. (Many of the same effects can be used on Speech Bubbles, only from a default shape of round.)

The simplest form is to depict something protruding from it in the Frame Break—but anything 2D is possible. There are a thousand ways to do this, and a million reasons.
 * To represent the passage of time.
 * A four-panel newspaper strip might go like this... First panel: Alice says something weird.; Second & Third panel combined into one double-sized panel: Alice and Bob stare at each other.; Last panel: Bob calls Alice a weirdo. (In this case, the long panel represents a long pause.)
 * A full page in a comic might depict someone jumping out a window. The page will be divided into narrow vertical panels depicting stages of the character's fall. (Here, the narrow panels indicate that the action is happening very quickly.)
 * Two events are happening simultaneously in different places. Two triangular panels joined along their hypotenuses depict the events.
 * "Pop-up" panels around a central illustration might depict different characters' reactions.
 * To visually reinforce the action.
 * An explosion might be depicted in a jagged, pointy... umm... explosion-shaped panel.
 * If broken glass is involved, the panels might be shaped like jagged fragments of broken glass.
 * The path of a bullet might be drawn in a panel which spans the page left-to-right, but only occupies a small amount of vertical space.
 * A character picking a lock might be depicted in a key-shaped panel.
 * To reinforce a character's thoughts, the theme of a conversation, or a theme of the work in general.
 * A character who is very angry might be drawn in... err... a jagged, explosion-shaped panel.
 * Two characters flirting might be drawn in a heart-shaped panel.
 * A situation involving recursion or infinite regress might be drawn as panels within panels, getting smaller to the point of invisibility.
 * To break up visual monotony. This is at least part of the reason for probably 95% of oddly shaped panels.

Depending on how they are juxtaposed, can make following the sequence difficult, since there may not be a left-to-right (or right-to-left in manga), top-to-bottom order. Sometimes panels are even "superimposed" as if they were on top of each other; this is commonest in the Sub-Trope Speech Bubbles Interruption, where it is used to show talking over each other.

Comics

 * Will Eisner's The Spirit may be the Ur Example of using unusual page layouts to visually reinforce story elements. This series invented many of the techniques mentioned above; indeed, much of the visual vocabulary of action-oriented comics can be traced to The Spirit. Only Orson Welles' film Citizen Kane is comparable in its acknowledged influence over an entire medium.
 * Jack Cole, who assisted Eisner on the Spirit for a while, would employ odd panels in his own comics, for example in the second Plastic Man story in this post all the parts of the story set in dreamland had wavy panel borders with black gutters in between them.
 * Sam Kieth's The Maxx did this all the time.
 * Frequently used in the Calvin and Hobbes Sunday strips, particularly after Bill Watterson's first sabbatical.
 * In the Tenth Anniversary collection Watterson says he spent a lot of time trying to escape the "tyranny" of panels.
 * Archie Comics did this routinely in the 1970s.
 * The first couple of issues of Elf Quest drawn by Wendy Pini almost exclusively used rectangular panels, but by issue #4 she was starting to experiment with more ambitious layouts.
 * The Death of Superman comics, in 1992, has a very interesting example. It starts by the wake of Doomsday, who then starts attacking everything in sight, until Superman arrives. Then Doomsday focus on him only, and they start fighting, without dialogue (since Doomsday can't talk at all and won't stop its attacks), each page having 8 panels. The next issue was more pure fight, with each page having 7 panels. Then 6, then 5, and so on. The last issue, then, is composed only of single-panel pages of Supes and Dooms beating the crap out of each other, and in the last one both of them die.
 * When he draws Detective Comics, JH Williams III divides the Batwoman segments from the ones focusing on Kate Kane by giving Kate standard panel layouts while Batwoman's scenes feature all manner of Odd Shaped Panels, from jagged-edged starbursts to fight scenes shown entirely in panels shaped like lightning bolts.
 * Frank Quitely often experiments with odd panels, for example he will occasionally make the panel the literal fourth wall of a room. In WE 3 he gets really inventive with sequences using a large panel with a series of tiny panels showing all the small details of the scene layered on top of the larger panel, or in one sequence he tilts a series of panels sideways as a character is moving through them, and its awesome looking.
 * In the Graphic Novel Joker, a crash involving The Joker and the protagonist shows the characters' reactions to the impact drawn inside of the word "CRASH" as if the word was a panel. It also uses the "Broken Glass" effect mentioned above, as if the reader is watching the characters' reaction through the breaking windshield of the car.
 * Sounds as panels also appear in Frank Miller's work, for example when Marv shoots a corrupt preist in Sin City.
 * Winsor McCay's Little Nemo in Slumberland (1905) and George Herriman's Krazy Kat (1914), are possibly the trope makers. McCay created the elastic layout- in the later pages, almost no two panels are the same shape. In several story arcs, he has panel borders break or shatter after being pushed- or in one case eaten by the main cast. He was also one of the first to use the 'explosion' panel. Herriman's sunday layouts featured nested panels, inset panels on open backgrounds, slanted and sliding panels, and circular panels, among others. The layout varied wildly week to week- except for a brief color run in the 20's- and sometimes dispensed with panels entirely.
 * The Sandman uses this often and to great effect, being an account of the Lord of Dreams and those connected to him.
 * Regularly used in Alan Moore's Promethea. Most often this trope takes the form of having the panels evoke mystic symbols relevant to the subject of the chapter.
 * One issue of John Byrne's run on Alpha Flight featured Snowbird facing one of the Great Beasts, Komolaq, described as "the living embodiment of winter". The battle featured a number of unusual frame shapes and placements, made all the more noticeable in that the frames were all blank(the idea was to depict the battle taking place within a severe blizzard, with nothing but flying snow visible). Only the frame shapes and locations, along with dialogue, provided any clues to the action. A Crowning Moment of Awesome for John Byrne, in that he made the fight work on the page.

Film

 * Ang Lee used these in his 2003 adaptation of The Incredible Hulk.

Manga

 * There's some panels in Ichigo Mashimaro that are convex quadrilaterals, some of which are right-angled trapezia. ...trapeziums? Nothing odder than that, though...
 * Bleach in particular uses this a lot, with fans of triangular frames to show individual reactions of members of a group.
 * Fruits Basket: The most common is a diagonal side.
 * Majin Tantei Nougami Neuro plays with this quite a lot.
 * JoJo's Bizarre Adventure uses a lot of slanted panels. The end of Part 5 gets especially hectic.
 * Ode to Kirihito contains spiral-shaped panels at points to convey multiple actions in quick succession by a single character.

Web Comics

 * Project 0 does this all the time to the point that no two pages are really designed alike. Notable example on page 6 where Owen's fall is fragmented into vertical frames like the symbol for signal strength bars to show that his powers don't 'get reception' in the Machine Graveyard.
 * No Rest for The Wicked has panels in the form of feathers.
 * Sluggy Freelance switches from its usual rectangular panels to slightly irregular, smooth-sided ones when depicting Gwynn's demon-possessed dream sequences. These are also surrounded by border text, usually the repetition of Arc Words, but occasionally the panel boundaries open slightly and the border text nearby changes to reveal a meta-clue.
 * Sen Diaz doesn't seem to like even rectangular layouts, and frequently uses circles or cuts the page into triangular shapes.
 * Casey and Andy did this once when they were literally breaking their own reality.
 * Tally Road constantly does it, either with radical and strange panel shapes or by simply using the angle of vertical panel borders to imply the state of mind of the characters. More askew angles are associated with more unstable feelings. At one point, passing time is represented by the panel borders 'sweeping' to increasing angles like the hands of a clock.
 * PolkOut has used some funky panels before.
 * HERO regularly uses oddly shaped or oddly placed panels for narrative effect.
 * Darths and Droids uses irregular polygon shaped panels whenever the action includes a fight sequence (usually with laser-swords).
 * El Goonish Shive has it well established that rounded corners are fantasy, dream, or narrative. The artist also likes to let force effect the panels, one particularly powerful punch actually made the whole panel order bow to the left.
 * Girl Genius. Examples here, here, and here.
 * Not just oddly shaped panels. Phil Foglio's many tools include overlays in sequence or to indicate relationships (or here and here) and perspective that would be done with sequential shots in cinema. Another use, to similar effect. He also carries 'effects' noises and speech bubbles across panels.
 * Murry Purry Fresh and Furry uses free-form panels with wavy borders.
 * The panels of Hanna Is Not a Boy's Name are very frequently irregularly-shaped (check out the center bottom panel here) or overlapping. It fits with the high-energy feel of the comic and its manic protagonist.
 * An even more dramatic example comes when Hanna counts to three: 1, 2, 3.
 * The Life of Nob T. Mouse often has the top of a character's head popping out of a skewed ellipse during conversation pieces.
 * Every panel in Cwynhild's Loom has rounded corners.
 * All Over the House also often has the top of a talking character's head popping out of an elliptical frame.
 * Gunnerkrigg Court: Here, and a trapezoidal one here. The comic very often skews the panel borders during action scenes. Etheric views usually have non-typical panel appearance, such as all melded together with objects (usually Annie's hair) serving as visual separators, or one panel as a background and others on top of it.
 * In Penny and Aggie, when Brandi knocks out Xena, the panel borders buckle as if from the force of the blow.
 * A Lesson Is Learned but The Damage Is Irreversible is famous for doing this effectively in a great many of their comics to reinforce events or theme, such as here.
 * Stuff protrudes from panels in The Way of the Metagamer a lot, but this comic is a particularly notable example.
 * Little Worlds regularly uses blurred and oddly shaped panels in its pages.
 * Pumpkin Flower normally has boxes in lines of three, but panels get funky when characters are hallucinating
 * Far Out There lives by this trope, seriously. The most jarring thing it can do now is a normal layout.
 * Keychain of Creation uses angled panel gutters for action scenes.
 * The Dreamland Chronicles, mostly overlapping.
 * Bob and George, rarely.
 * overlapping here
 * Sliced into triagles to show simultaneity
 * Megatokyo: trapezoids
 * Non-rectangular panels are rare, but more recently, floaty-freeform-layout panels superimposed on a partial background have become the norm. Tilted panels show up occasionally in action scenes, such as the "Kyaaa!" panel here.
 * Last Res0rt has one page where one of the characters goes out of the panel(possibly onto the next page)
 * Wapsi Square like here
 * Pibgorn triangles
 * Eerie Cuties pentagon
 * Quentyn Quinn, Space Ranger: Three panels merged into one -- one on top, two below
 * Evil Diva: overlapping here and here
 * Wooden Rose not lined up straight or with slanted edges
 * Fey Winds Slanted and overlapping
 * The Phoenix Requiem Slants here and here.
 * Actually, odd shaped panels are the norm in this comic rather than an exception.
 * Strays slanted sides and superimposed
 * Familiar Ground An inset square
 * Dork Tower Cut-out corner
 * And arched sky
 * Tamuran slanting sides
 * Fuzzy-edged
 * Footloose Slanting sides
 * Impure Blood superimpose a long, short panel over the middle division
 * Dreamkeepers Prelude
 * Warped squares, with nicks
 * Warped sides, and ovals, and without borders.
 * Larger and smaller for conversation across a door, to indicate inside and outside.
 * Little scenes overlapping the large one
 * Roza
 * One panel slips down to overlap
 * slanted sides
 * Thistil Mistil Kistil During a fight, slanted sides
 * Derelict as wide as the strip, and wavey, to indicate that she slept a long time.
 * In Nip and Tuck, one without a border, to indicate it's not part of the Flash Back.
 * Again, with the border as a "thought" shape
 * In Endstone, slantwise borders during a fight.
 * In Next Town Over, framing faces in panels like lockets.
 * In Doodze, slanted across the line of their fall
 * In American Barbarian, a triangle to introduce the second plot line.
 * In The Adventures of Shan Shan, superimposed and trapezoidal.
 * In Blue Yonder, superimposed.
 * This Order of the Stick strip. Thog lampshades this in a much later strip.
 * In Erstwhile, Maid Maleen's inability to find work is showing in a sequence of round panels.
 * In Far From Home, Eyak's reaction to the pilotting is shown in a hexagon on top of two panels showing the ship dodging asteroids.
 * In Ears for Elves, a series of short, wide rectangular panels indicate zooming in/panning down here.