Space Navy

As we all know, Space Is an Ocean. So naturally, what does The Federation do when it needs someone to defend it in space? Why, it turns to its Space Navy. A Space Navy has a Standard Sci-Fi Fleet naturally enough, which probably includes at least one class of Cool Starship. Likely it will have an arm of space marines. But it has more. It has the same sort of atmosphere of a real navy. It has organization and obstructive bureaucrats. It has tradition and famous names. It might even have in-jokes, perhaps the same ones as when a given writer was in the navy (apparently these are very old jokes by now). Naturally the heroes will likely be part of a Command Roster, for this is what they have on a Space Navy.

Anime and Manga

Legend of the Galactic Heroes
Star Blazers. Here noted for the fact that it draws at least some inspiration from the Imperial Japanese Navy and is therefore an exception to the common usage of basing it around the American and/or British Navies.
- On the other hand both the American and Imperial Japanese navies were borrowed a lot from the Royal Navy. To be the best one must learn from the best.
Crest of the Stars, though it become more prominent in the sequels.
Irresponsible Captain Tylor
Gundam has the Earth Federation Space Force.
UN Spacy, and later, the NUNS.
In Magical Girl Lyrical Nanoha, the TSAB has an interdimensional navy.

Film

Star Wars: The Rebel Fleet and Imperial Navy in the Original Trilogy. Though the heroes were more outside the power-structure of just the Rebel Fleet, being able to function in different capacities. In the Star Wars Expanded Universe, Ascended Extra Wedge Antilles has his adventures in a pure Navy-setting. Interestingly, while it would appear that fighter units use naval ranks (Luke is mentioned as a Commander), there are also several generals, implying a separate army.
Wing Commander: The film, basically being World War II In Space, centers around a carrier, the TCS Tiger Claw, one of the few surviving ships of a Kilrathi attack on their home base. They must delay the enemy long enough for Admiral Tolwyn to move his fleet into position to defend Earth.

Live Action TV

Star Trek: Seemingly a 50-50 mix of the Age of Sail and World War II navies.
Firefly: The Alliance fleet. Usually a nuisance.
The Colonial Fleet, or what's left of it, in both Battlestar Galactica series.
- In the reimagined series, the concept is actually deconstructed to a point. Pre-Cylon attack, the trope is played deadly straight, but once the series starts the "fleet" has been reduced to a single military Cool Ship (and the requisite fighters aboard). Scenes on the Battlestar are still played straight, but in general the series has a very different tone than most space navy shows.
The Daedalus-class battlecruisers operated by the United States in the Stargate Verse both averts it and plays it straight. On one hand, the ship is operated by the United States Air Force, and the rank structure is the same (so the equivalent of The Captain is in fact a colonel). On the other hand, the ships have the same "USS" prefix as commissioned Naval ships, although Fanon held for a while that the prefix was "USAFV" (United States Air Force Vessel, by analogy with USAV, United States Army Vessel; "USAFS", the actual designation for a USAF ship--not currently in use--was apparently not cool enough).
Babylon 5 most prominently featured EarthForce, which most (but not all) of the human characters were members of until mid-Season 3.
Space: Above and Beyond featured the US Navy in this role, along with the implied participation of several other nations' space navies in Earth's war against the Chigs. We see a few representatives of the other militaries, but the Americans take the center role in the show (given that the show centers around an American Marine Corps squadron, this is Justified.)
The old Systems Commonwealth in Andromeda had the High Guard, although the titular massive warship often ends up behaving like a jet fighter with barrel rolls and other improbable maneuvers. Like Starfleet, the High Guard is a combined service, with the main two branches being the Argosy (starship officers) and the Lancer Corps (ground troops and Space Marines). Unusual for this setting, there is plenty of Interservice Rivalry between these two groups, although the only thing we see in the show are some nicknames being thrown around.

Literature

The Galactic Patrol in E. E. "Doc" Smith's Lensman novels, when they were fighting the space fleet of Boskone.
The Warchild Series has a Space Navy that was pretty much called just that.
An Exchange of Hostages and its sequels deal with a medic/torturer assigned to the resident Space Navy.
The entire Honor Harrington series is about life in one of these, lovingly detailed.
The Seafort Saga is set within the United Nations Naval Service. An actual Space Navy.
In Larry Niven's story "The Return Of William Proxmire", one of the changes to the present caused by Time Travel and Robert Heinlein not being discharged from the Navy due to tuberculous is that space exploration is now part of the Navy's brief, led by Admiral Heinlein.
Larry Niven's Known Space also has the UN Space Navy, particularly involved in the Man-Kzin war series.
Robert Heinlein's Starship Troopers uses naval terminology whenever military spacecraft are involved. The characters even refer to seagoing military ships as a "wet navy" to distinguish them from the space navy they're more familiar with.
The Kris Longknife series.
Vorkosigan Saga has a few references to the Barrayaran navy. It seems more like that of Imperial Russia or Imperial Germany. naturally.

Tabletop RPG

Spelljammer Dungeons and Dragons campaign setting has Imperial Elven Navy infested by obstructive bureaucrats vs. hodgepodge Scro fleet as major forces; Shou and some other groundling empires has their own fleets too.
There are several examples of a Space Navy in Traveller.
The Warhammer 40000 spin-off Battlefleet Gothic is all about this, and despite being set in the 41st Millennium has a very Age of Sail feel to things - tactics concern broadsides, line formations, and crossing the enemy's T, and the hordes of crewmen manually loading the plasma cannons were probably Press-Ganged from the last planet the ship visited.
Partly averted in Transhuman Space, where the USA's military space presence is the ambit of the Air Force, and thus many traditional "ships in space" tropes are averted.

Video Games

Wing Commander has not only a Space Navy, but an Air Force analog (Terran Confederation Space Force), as well. There's no real rhyme or reason as to when a carrier is host to a Navy wing or a Space Force one, and there's apparently some switching of personnel between the services (Commodore [a navy rank] Blair in Prophecy was, prior to 2681, in Space Force, which uses a modified Army rank structure). The ships themselves, however, are pure navy, and manned by navy crews.
- There's an element of Truth in Television to this, as some countries here on Earth have been known to go back and forth on whether their naval aviation assets should be operated by the Navy proper, seconded from the Air Force or a separate entity altogether. Britain in particular has a tradition of this sort of thing stretching back nearly as long as there have been British military aeroplanes; see The UK Armed Forces for details.
The Systems Alliance navy from Mass Effect.
SolForce in Sword of the Stars. It's also the de facto government of humanity, and the director of SolForce is the closest thing to a secular leader humanity has.
Halo features the United Nations Space Command.
Free Space has the Galactic Terran Alliance and the Parliamentary Vasudan Navy. In the second game, they merge into the Galactic Terran-Vasudan Alliance.

Web Comic

Taken to rather literal extremes in Pockett, with crew members dressed in modern looking navy uniforms.

Web Original

Star Army is a universe full of space navies. Almost every nation has one and they're one of the most detailed parts of the military sci-fi setting.

Real Life

So far averted, what few space based military operations exist currently (mostly spy satellites) are under the Air Force's jurisdiction.