Homeless Pigeon Person

A homeless person with a loyal animal companion. Usually a clear signifier that they're a good guy and not one of the Crazy Homeless People. Note that this refers to all kinds of animal companions among the homeless. Pigeons, for whatever reason, are the most common.

Interestingly, even though pigeons are generally considered "flying rats" by the populace at large (some cities have regulations against feeding them), most characters with pigeon friends are portrayed in a favorable light. Probably has something to do with helping the helpless.

Crazy Cat Lady is a far more negative portrayal, though cats are generally agreed to be fluffy and cute. Also see Crazy Homeless People.

Examples of Homeless Pigeon Person include:

Comics

In the former alt-newspaper comic strip Phoebe And The Pigeon People Phoebe was originally homeless.
The homeless man in WE 3.

Film

That homeless pigeon-caretaker woman that Kevin McAllister befriends in Home Alone 2.
The Bird Woman in Mary Poppins.
While he's not homeless, (if only because he lives in a one room shack on a roof) the lead character in Ghost Dog becomes known to the bad guys as 'the ****ing bird man' cause he keeps pigeons.
Subversion: Franz Liebkind, the crazy ~~ex-~~Nazi playwright from The Producers. Here, the pigeons prove that he is one of the Crazy Homeless People beyond a reasonable doubt. Mainly because he uses them for his crazy Nazi hijinx.

Literature

The title character from Wizard Of The Pigeons, an Urban Fantasy novel by Megan Lindholm.
Mrs Tachyon, the time-travelling bag lady from Terry Pratchett's Johnny Maxwell Trilogy, has a pet cat (only the one, so she avoids being a Crazy Cat Lady through a technicality). Mrs Tachyon is a fairly benevolent, though totally incoherent, character; the cat is pure evil.
Old Bailey from Neverwhere.
The protagonist of Like Water for Chocolate, Tita, goes through this phase after her mother sends her sister Rosaura and Pedro away, a move that results in the death of Tita's nephew, whom she had been breastfeeding because Rosaura had been unable to feed him. In her grief, Tita takes to living in the dove-cote and treating the doves as children.
The Duck Man in Discworld appears to be an example of this trope. However, conversation will reveal that he is, or claims to be, completely unaware of the duck that gives him his name, despite the fact it's on his head.
Merivel in Restoration by Rose Tremain could be seen as this by the rather non-sequitur end of the story - he has lost much, but at least the King has given him some pigeons in the house he once occupied.

Live Action Television

Claude from Heroes. He basically walks a fine line between this and Crazy Homeless Person.
There was a one-shot character on Cold Case who had known and mentored the victim of the week and was your typical pigeon keeper. He's like this due to depression caused by a mistake in his airplane design leading to several accidental deaths.
Radek Zelenka from Stargate Atlantis reveals he's a major pigeon enthusiast and used to raise racing pigeons back on Earth.

Video Games

Beggar Crazy Clara in Dreamfall has a pet that denies biological definition and that Zoe has to rescue from Azadi to get Clara to tell her some plot-relevant stuff.
Senor Cardgage is mentioned to be one of these in Strong Bads Cool Game for Attractive People: 8-Bit Is Enough after Trogdor goes on a rampage.

Strong Bad: You can usually find Senor Cardgage reading to the pigeons behind here, but now there's just all these burnt chicken bones. I guess Trogdor must've scared them-- Ohhh...

Web Comics

The aptly named Pigeon Man from Transient Man. Not a heroic example, though. Most definitely not. Might better fit the description of Homicidal Pigeon Person.

Western Animation

Hey Arnold! has a "pigeon man" (pictured above) living on a rooftop, with some elaborate cages containing hundreds of pigeons. At the end of the episode, he actually flies away carried by his birds.
In Disney's Oliver and Company Fagin is a homeless guy in New York who lives with a pack of dogs, and eventually takes Oliver the kitten in.

Real Life

Truth in Television: Near the end of his life, Nikola Tesla took to moving between hotels and keeping large flocks of pigeons.
Mike Tyson, of all people.
Run-down houses sometimes have pigeons roosting in their attics. Major health hazard. Why? Their excrement hardens into a fragile powder that, once disturbed, will hang in the air like dust and can be inhaled. So now you know.