Bizarre Seasons

This is a trope where rather than have the normal Spring, Summer, Fall, Winter progression in a nice predictable order, the setting has seasons that are weird. It may have extra seasons, or seasons with strange weather, or perhaps the seasons change randomly.

Note this is for works that have this as an explicit part of the setting, it doesn't count if it's just the result of writers and producers either being bad at keeping track of what season it's supposed to be, assuming It's Always Spring unless the weather is plot-relevant, or just not giving a darn. It also doesn't count if the craziness was imposed by a deliberate magical spell or other similar effect, unless it happened so long ago that the characters don't think anything of it.

Examples of Bizarre Seasons include:

Anime and Manga

The Grand Line of One Piece is filled with with bizarre weather between islands. From clear to snow to rain in minutes. In fact, the weather stabilizing is a sign that you're near an island. As for the islands themselves, localized weather patterns can result in islands that are a single season for much of the year, giving us "Winter Islands" or "Summer Islands." This creates a minimum range of 16 potential weather patterns as you have the extremes of winter on a Winter Island to summer on a Summer Island. And that's not even going into islands where you have things such as constant rains of lightning.
In Neon Genesis Evangelion, the Earth's axial tilt was shifted during Second Impact. The result of this is that It's Always Spring.

Film

Monty Python and the Holy Grail. During an animation:

"Winter changed into spring. Spring changed into summer. Summer changed back into winter. And winter gave spring and summer a miss and went straight into autumn."

Literature

In Larry Niven's Known Space series, several of the planets have odd seasonal patterns. During winter on We Made It, the planet's northern axis points directly at its sun, causing hundred-mile-an-hour winds on the surface (the inhabitants of the planet live underground). Summer and Winter on Shast are identical in temperature, but summer is much, much wetter. And what's unusual about the seasons on the Ringworld is that there are no seasons. Period. Its pretty much always summer there.
The Stormlight Archive is set in a land where seasons change apparently randomly every few weeks, with the only predictable weather being a few weeks of constant rain that mark the end of one year and the beginning of the next.
Discworld has the normal 4 seasons... each of which occur twice a year. There's also a decent chance that April is an autumn month on the Discworld, though that may be just a continuity glitch.
In A Song of Ice and Fire, seasons last years, especially summer and winter. The beginning of the series is the end of one of the longest summers in recorded history, which means winter will be even longer. Needless to say, Winter is Coming.
The Honor Harrington series of books occassionally go into detail on the weather of various planets, since none of them have the exact same length of years or seasons as Earth does. Honor's homeworld of Sphinx has 15 months, with the seasons being four or five months long, with some noteworthy climate extremes. Many of the buildings have solid walls and very tall, steep rooves to keep the tremendous amounts of snow from piling up and crushing the buildings.
A Doctor Seuss book had wizards making a new kind of weather.

Tabletop Games

Dungeons & Dragons adventure OA7 Test of the Samurai. The partial plane of Qui has four seasons: Phoenix, Turtle, Unicorn and Dragon. Each has significant effects on the weather and terrain, including changing water to lava (Phoenix), constant rain (Turtle), almost normal (Unicorn) and heavy fog (Dragon).
Magic: The Gathering has Lorwyn and Shadowmoor. Every hundred years, Lorwyn goes through one "day". Lorwyn is the "day" time and Shadowmoor is the "night" time. Flavor-wise, most races are at their best in Lorwyn, and, except for the elves, it's a total Sugar Bowl. Shadowmoor, however, has the worst aspects of the races. The aforementioned elves, and the faeries who are behind the whole thing, are the only ones immune to it. And nobody remembers Lorwyn when Shadowmoor returns. Mechanically, Shadowmoor is bizarre: Removing weakening counters to do things, enemy-color associations, and untap costs.
The RuneQuest world of Glorantha has five eight-week seasons, corresponding to its five elements: spring-like Sea Season (Water), summer-like Fire Season, autumn-like Earth Season, winter-like Dark Season (Darkness), and turbulent Storm Season (Air). There's also two weeks of Sacred Time between Storm and Sea, spent in performing death and rebirth rituals and seeking oracles of the year to come.
In Swashbucklers of the Seven Skies, floating islands experience six seasons over the course of a 360 day year, as each of six skies passes overhead: The Mists are warm and foggy, The Jungle Sky is even warmer and filled with floating trees, The Sky of Thunder is full of rain and lightning, the Sky of Stones marks early autumn, the eerie and colder Ghost Sky marks late autumn, and the Sky of Frost is icy and nearly impassible. (The seventh sky, the Sky of Fire, sits at the central axis of the world dome and fries anyone who tries to enter.) Ships travelling far can cross into a sky earlier or later, and ships bear the brunt of each sky's weather, notably stones, lighting, and trees.

Video Games

In Borderlands, seasons in Pandora are long. As in almost a decade long apiece. When the big conglomerates colonized the planet, they made the mistake of arriving during the winter cycle. Seven years later, spring came, which meant mating season for the planet's fauna. The conglomerates have since basically given up on the planet, considering the incredible hassle the sudden infestation of skags, rakks, sylithids and other beasties has caused.
The plot of The Legend of Zelda Oracle of Seasons is that the titular Oracle of Seasons has been kidnapped, and without her, the seasons have been thrown into chaos. Winter, spring, summer, and fall occur all at the same time in different areas, sometimes randomly switching, and the player character eventually gains the ability to manipulate the seasons at will.
In The Sims 2, with the "Seasons" expansion, you have four slots which each can hold any of the four seasons. Bizarre combinations and inexplicable orders are very doable.
In Pokémon Black and White, the season changes every month, so it could very easily be autumn in the middle of March.
Kirby Super Star has planet Floria, where the seasons change whenever you enter a door.
In Final Fantasy XII, one area has a wet season and a dry season. This doesn't change until the plot demands, and it will change back when the plot allows, but after that, it's set on a cycle. Definitely not an annual one, either.

Web Comics

In Off White, the winter has lasted for over three centuries for some unexplained reason.

Western Animation

In South Park, there are two seasons: Winter and July. Also, even the summer has snow.
One episode of Rugrats had all four seasons happen in a week.

Real Life

Seasons as based in weather can be very different and very weird in some parts of the world. The most common cycle besides the European 4-seasons system is a 2-season system, with a wet season and a dry season. This applies primarily to deserts and Southeast Asian countries.
- In Antarctica, the primary difference between "Summer" and "Winter" is how much sunlight the continent receives, and how bad the storms are. Same in the arctic.
- The inhabitants of Northern Australia, at least around Kakadu, where the weather is dominated by the monsoonal pattern recognised six seasons based on the intensity of the rains for the 'wet' half of the cycle and how much free water was still around in the 'dry' half.
Spaniards of Madrid and south claim to have "9 meses de invierno, 3 meses de infierno"
El Niño is one of many reasons agriculture never developed in Australia.
Parts of Texas are renowned for this. Particularly in "Spring" and "Fall," it's not unheard of to have a 30 or even 40 degree differenced between one day and the next.
In India, the four seasons are Winter followed by Summer, followed by Monsoon followed by Post-Monsoon.
The planet Uranus actually spins on its side, so one pole actually remains in constant darkness for almost half a year.
- That's half a Uranian year, which is around 42 terrestrial years.
- In a similar vein, if a planet orbits one star in a binary system, then during part of the year the more distant star will be on the opposite side of the sky from the closer one, resulting in several extra hours of (much dimmer) daylight.
There are plenty of jokes about the weather in one's home region (or sometimes another place that the speaker doesn't like) falling into a set of less-than-pleasant esoteric seasons. Seasons like Rain, Fucking Hot, and Road Construction.
- In New England, early spring is known as "Mud Season." However, the weather tends to be so erratic in New England that at times the label "seasons" seems to be a stretch. Even in winter, it can go from -10 degrees and blizzard conditions one day to 45 degress and sunny. In the summer, it's not unusual to use both your air conditioning and your heater in the same day.
  - A common tourist t-shirt in Vermont Lampshades this with their "weather report": "Hot and sunny, turning to sleet mid-morning, becoming a blizzard by noon, 6-42 inches, followed by warm, moist, tropical air that will probably bring severe thunderstorms and a floodwatch."
- In Florida, you've got Lovebug Season, Tourist Season, Hurricane Season, and February.
- In California, you've got Riot Season, Fire Season, Flood Season, and Pilot Season.
- In Western Oregon (this could also apply to Western Washington and coastal British Columbia) there's Rain, Rain, Road Construction, and Rain. Oregon's Eastern side, on the other hand, is a desert that gets lots of snow in the winter.
- Rio has two seasons - Summer and Hell.
  - No, no we don't. Summer can be up to the nineties, and Winter can be at fifties. Winter is a little bit colder, Summer is a little bit warmer. Otherwise every other day can vary greatly. Except for June Gloom.
- In Croatia the names of months are: Chopping wood, Changeable, Cat season, Grassy, Dogwood, Lime, Sickle, Driving cars, Ruddy, Falling leafs, Cold and Begging. I kid you not.
  - Lime is still better than Small Red Bug. (More at this search.)
- In Minnesota the seasons are known as Winter, Mud, Road Construction and Still Road Construction
  - Or, sometimes, Snow Removal and Road Repair. Possibly the other way around.
- Central Canada has Almost Winter, Winter, Still Winter, and Construction.
  - Arctic Canada has Night's Here, Getting Dark and Cold, Cold but no Snow, First Snowmobile Season, Too Damn Cold to Snowmobile, Second Snowmobile Season, Mud Season, and Summer. Summer is usually a Tuesday.
- South Dakota has chilly, blizzard, flooded, and hotter than hell.
- Texas: if you don't like the (current) weather, just wait five minutes. Or (in 2011) two months.
  - Ditto for Colorado.
    - Or pretty much every state except Hawaii.
- How do you know it's summer in Wales? The rain's warmer.
- In Michigan, we have have two seasons: winter and construction season. They often overlap.
  - Same with Wisconsin.
- Here in Arizona we have Summer and Slightly cooler Summer (Lows in the 'winter' of 50 F and highs of 120 F in Summer, normally.)
- Melbourne, Australia is often joked to go through all four seasons in one day. Days have been known to swing from the high 30s and low 40s (90-100 for those using Farenheit) during the day to below freezing at night. (This is an extreme example but it does happen)