Future Copter

You're in an unspecified future, and you need an aircraft capable of short or vertical takeoffs. Helicopters are so 20th century: they can't do all that fancy supercruising and barrel rolls that jets can do and they (probably) can't survive being dropped from orbit because well, hey, you might just need to be dropped from orbit.

What you need is a futurecopter. These machines come in a variety of sizes, ranging from dinky sized models about the size of an RC toy, a one-man unit (with optional handholds on the sides for external passengers), through unmanned autonomous versions (best steer well clear of these) the size of a Hummer, right up to beasts the size of a civil cruiser capable of transporting dozens of fully armed troops complete with an airlifted Awesome Personnel Carrier or a Cool Tank. Size and operation aside, what they almost invariably have is ducted fans (one each side of the fuselage, and usually independently tiltable) for lift and thrust, a V-tail and stubby, vestigial wings with an assortment of guns, missiles and bombs, if there were any wings at all. And, in compliance to the all-powerful Rule of Cool, you can expect the aircraft to be as fast as a jet, as slick as a helicopter and as armed as a fighter-bomber. The pilot, if present, is invariably right up front in what looks like a conventional helicopter gunship cockpit.

There are currently no production manned aircraft that look anything like this, for sound aerodynamic and engineering reasons which are unrelated to the Rule of Cool. They are being developed, however, and at the time of writing (2011) probably the nearest to production is the Urban Aeronautics Airmule. One might, with some imagination, consider existing tilt-rotor aircraft like the V-22 Osprey to be a very primitive version of this idea, although its rotors/propellers are firmly locked together and very much not ducted, and it has big, conventional wings.

For the one-man version, see Helicopter Pack.

Examples of Future Copter include:

Anime and Manga

The JSSDF VTOLs from Neon Genesis Evangelion have tilting jet engines.
Vehicles like this are seen in the first episode of Bubblegum Crisis
The Tiltrotors in service for Section Nine, as well as the requisite black choppers.

Film

The 1930's sci-fi musical Just Imagine, everyone has a personal airplane with vertical rotor ducts in the wings enabling them to hover while they talk to other drivers or burst into song.
The human forces in Avatar have helicopter sized ducted-fan gunships.
The Hunter-Killer craft from Terminator and Terminator 2 are of the ducted-fan VTOL type. (The first movie may be the design's first appearance in any media). Terminator 3: Rise of the Machines features small, autonomous prototypes which pursue the heroes.
In Aliens, the Colonial Marines' dropship is a fully spaceworthy version of the design. The rotors on a Hornet aren't attached to wings, and the Pelican Drop Ship and the Aliens Colonial Marine dropship don't have tiltrotors at all, they're still spiritually close to this design: The critical features are VTOL flight envelope, V tail (probably), gunship-canopy pilot position... and not much else. Fundamentally they serve the narrative function of a helicopter - hardly surprising given the Vietnam War parallels that are sometimes seen in the movie.
The Incredibles. Flown by the mooks guarding the Supervillain Lair.
Arnold Schwarzenegger plays a future helicopter pilot in The 6th Day. Naturally, the copters he flies are Futurecopters, with the ability to switch to a high speed jet mode and be flown remotely using a Power Glove-like arrangement.
A.I.: Artificial Intelligence: the amphibicopter notable not only for If It Swims, It Flies but for lacking rotors while still serving as a helicopter.

Live Action TV

Big Bad Beetleborgs's ladybug aircraft also fit the trope.
Super Sentai has had a few, starting with Rescue Sentai Go Go Five's Green Hover, as well as the Pat-Gyrer from Tokusou Sentai Dekaranger (whose cowlings form a set of Humongous Mecha-scale handcuffs when it combines into Dekaranger Robo) and the GouGou Gyro in Gou Gou Sentai Boukenger.
The ornithopters in the 2000s miniseries adaptation of Dune and its sequel appear to be this. While their wings may move up and down a little to adjust pitch, they don't flap like the 'thopters in the book and are more bug-like than bird-like in appearance. The miniseries' carryalls are of the Future Copter variety as well, complete with prominent ducted fans mounted on four extrusions from the hull of the craft.

Literature

Several different versions (from supersonic gunships to massive transports) are seen in the War Against the Chtorr series, often piloted by the protagonist's Love Interest, Colonel Badass Lizard Tirelli.

Tabletop games

Nurglite blight drones from Warhammer 40,000.
Generic RPG supplement Booty and the Beasts. The Whirly Chair is exactly what it says: a chair with overhead helicopter rotors. It has a maximum speed of 30 m.p.h. and a range of 300 miles on a tank of alcohol. It seats one person and has a small cargo area, enough for a small overnight bag or suitcase.

Toys

Mega Force (Kenner Toys)
Many newer (post-1990) LEGO sets with a futuristic theme.

Video Games

Halo 3: One-man aircraft called Hornets, which have external handles allowing other players to hitch a ride. The Pelican Drop Ship is a pretty close cousin of the dropship from Aliens, and is similarly orbital-drop capable.
- And in Halo Reach, the Hornet is replaced (or replaces given the continuity) by the Falcon, extremely similar to the V-22 Osprey but smaller, with LM Gs on the open sides and jet engines.
The GDI Orca and its variants from the Tiberium series of Command & Conquer are ducted-rotor VTOLs. Some variants go as far as being a Drop Ship.
- Nod counterparts eventually surfaced in the second half of the entire series and somehow, turned the tables on GDI in air superiority.
Battlefield 2142 has two futurecopters from two opposing sides of the conflict (the Talon for the EU and the Doragon for the PAC).
Enemy Territory Quake Wars
StarCraft II's Medivac dropship and Banshee ground attack vehicle use ducted fans.
One of the trailers for the Beyond Good and Evil sequel feature a craft like this.
The Chrysalis Pod in Metal Gear Solid Peace Walker is an automated drone example of this.
Combine gunships from Half-Life 2 have a single ducted fan.
Enclave Vertibirds in the Fallout series are clearly inspired by the V-22 Osprey.
The hovercopter in Perfect Dark, and Killian's VTOL jet transport/gunship in PD Zero.
Red Faction II has ducted fan gunships.

Western Animation

Global Justice field tilt-fan VTOL craft when they appear in Kim Possible, and when Ron briefly becomes a multi-millionaire, he buys an aircraft for Kim that resembles an Osprey with jets instead of rotors.
Cybertron-mode Megatron from Transformers Animated turns into a Cybertronic VTOL fighter for a split-second in the show's pilot episode. When he is rebuilt in his Earth mode at the end of Season 1, he now turns into yes, a modern V-22 Osprey helicopter.