Homeless Hero

There is the hero that drifts from place to place helping people as he walks the earth. Then there are these heroes: ones that stay in the same place, but are so down on their luck that they live in squalor or, worse, have no place to live at all.

Perhaps their dedication to doing heroic deeds provides them with little time for making money or starting a home or family. Or perhaps they've had a string of bad luck, sometimes due to a decidedly non-heroic addiction. In any case, even though they may save lives, normal people will still dislike them.

Examples of Homeless Hero include:

Anime and Manga

On Ranma ½ Ryoga may as well be one of these. Yes, he has a real house, but he's almost never there. In fact, the entire family is equally prone to being lost, so the house really only exists as a wayside stop. Most days he simply camps out, or sleeps in Akane's bed as her pet pig.
The title character of Rurouni Kenshin had been one for ten years before the official start of the story. Then Kaoru let him move into her dojo.
Sakuya from Dancougar Nova was homeless and unemployed before being recruited to pilot the titular Super Robot, and stays that way for a few episodes.
- Pointing out that he is, or was homeless is also something of a catchphrase for him. Gets a little ridiculous after a while.
The Satoshi Kon movie Tokyo Godfathers revolves around a trio of homeless heroes as they struggle to return an abandoned baby to her mother.

Comic Books

The new Doctor Fate from Steve Gerber's Countdown to Mystery was a successful psychiatrist until one of his patients went on a killing spree. He eventually ended up homeless and alcoholic - then the Helmet of Fate landed in his life. By the end of the mini, he'd at least found a job and somewhere to live.
Access from DC vs. Marvel (or vice-versa) not only is homeless, at least in his own future, he doesn't even have a home universe.
The indie comic Street Angel played this up with one issue of the book (which is normally about defeating government-issue ninjas or wacky threats like that) dedicated to the main character scrounging for food.
In our world, The Maxx literally lives in a cardboard box.
The titular character in Usagi Yojimbo, naturally, considering the fact that he's a Ronin
D-Man of the Marvel Universe (once sidekick to Captain America (comics)) was homeless due to a mental illness. Last time he was seen, he was getting treatment.
Watchmen's Rorschach lives in squalor and has poor personal hygiene.
Ragman from the DCU is also a hero who seems to have no home.
Marvel's Sub-Mariner spent the gap between The Golden Age of Comic Books and The Silver Age of Comic Books living as an amnesiac hobo. Johnny Storm found him and helped him remember who he was.
Tony Stark once hit rock-bottom, lost everything, and wound up a homeless alcoholic. But he also gave up being Iron Man for the time.
There was also a time in the 1960s where Captain America (comics) lived in a series of seedy hotels, before taking to the open road on his motorcycle. It made for an interesting metaphor about the state of country at the time, but it was more than a little annoying to read him bemoaning his lack of home and family, when all he had to do was join The Avengers again and move into their mansion.
The Crow wears found clothing (boots from a dumpster, etc.) and lives in an abandoned building. The fact that the apartment was his former home only adds dramtatic signifance to its current state of squalor.
The DCU's Uncle Sam was homeless for at least a little while.
Squadron Supreme vol. 3 also has a homeless incarnation of liberty (or something along those lines).
Then of course there's Top Ten, where everyone in the city has powers and a Code Name - and like any other city, there are homeless people.
The Punisher has to move pretty frequently among the safehouses he's established around the area. They are often abandoned buildings, storage units, or even little used access areas of the New York Subway.
Spawn - not many places where a literal demon would be welcome, but he doesn't really need housing.
Apollo and Midnighter spent many years after their escape from Henry Bendix living in abandoned warehouses and being damn grateful that they didn't have to eat or sleep much. They were still fighting crime, though, and it caught the attention of the new and improved Stormwatch, who helped them out. Then along comes Jenny Sparks with a proposition...
Flaming Carrot is a vagrant; when not righting wrongs, he doesn't do much except drink corn whiskey with other bums and read comic books.

Film

Hancock
The Fisher King (the Terry Gilliam film) fits this well.
Aladdin. Naturally, he shamelessly steals to live, but shares his bread with neighborhood children if compelled.
Nada from John Carpenter's They Live!

Literature

The Name of the Wind: Kvothe spends much of the book homeless in Tarbean.

Live-Action TV

An episode of Heroes had a homeless veteran that could turn his arm metal.
- Also on Heroes, Claude was an Invisible hobo. He stole everything he needed.
- Daphne too in her origin in the Graphic Novels.
Jim Rockford of The Rockford Files lives in a crappy little trailer.
Parodied on That Mitchell and Webb Look, with "The Surprising Adventures of Sir Digby Chicken-Caesar." Although he claims to be fighting an unknown nemesis "who is presumably responsible," in fact he and his sidekick are simply hobos who randomly attack people.

Newspaper Comics

Frank and Ernest often appear as bums in their strip.

Tabletop Games

This trope is a stereotype of heroic fantasy RPGs like Dungeons & Dragons to the point it's easier to list games where it's explicitly not the case. This tradition is often semi-affectionately known as "Murderhobo", so named for the tendency of stock heroes to wander from place to place, without permanent residence, killing things.
- Averted in most Living Campaigns, since significant amount of time is spent off-screen. In those characters generally have day jobs and off-screen homes by default.
While most clan samurai in Legend of the Five Rings have actual homes (however rarely they actually stay there), ronin player characters often fall into this.

Video Games

Ryu from Street Fighter travels around the world seeking opponents to challenge and has no home to call his own because of this.
Oracle of Tao. Subverted. The main character, Ambrosia, is a Crazy Homeless Person who cleans up a bit, losing the homeless status when she gets money from killing monsters and is given a tent. She no longer is homeless by the end, but arguably is still very much crazy.
Whether Gregory from Five Nights at Freddy's: Security Breach can be called a hero is debatable (it seems he was just in the wrong place at the wrong time), but he does seem to be homeless, the Bad Ending showing him sleeping in a cardboard box in a ally. More than likely, he snuck into the Pizzaplex hoping to steal food only to be trapped inside when it closed.

Web Comics

MAG-ISA -- Claudita doesn't have a home. Poor girl.

Western Animation

A variant occurred in Spider-Man: The Animated Series - one of Captain America (comics)'s old comrades spent the World War II living rough so he could secretly watch over the facility where Cap and the Red Skull were imprisoned.
By virtue of being on the run, the G.I. Joes are forced to become this trope in G.I. Joe: Renegades.
Wonder Woman in DC Super Hero Girls, due to a tradition stating an Amazon must live wherever she finds herself when traveling to a new land. The other Hero Girls try to remedy this, but Diana makes for a rather annoying guest. Eventually she is taken in by an archeologist studying Themysciran history.