After the events of Prisoner of Azkaban, Sirius spends the best part of a year (Goblet of Fire) living in a cave, and during the Order of the Phoenix lives in his family's house, Grimmauld Place.
It's pretty obvious that he hates Grimmauld Place, but Dumbledore made him stay there:
“Yeah, he did hate [the house]!” said Harry, his voice cracking, turning his back on Dumbledore and walking away. The sun was bright inside the room now, and the eyes of all the portraits followed him as he walked, without realizing what he was doing, without seeing the office at all. “You made him stay shut up in that house and he hated it, that’s why he wanted to get out last night—” — Order of the Phoenix, chapter 37
At the beginning of the book, he transforms into Padfoot to go and see Harry off at the station, but this is considered dangerous, as the Death Eaters know of his disguise from Pettigrew, and the Ministry is still using him as a scapegoat when things go wrong (e.g. the Death Eaters escape from Azkaban).
Granted, he is also less useful than his Order of the Phoenix peers, who had jobs at the Ministry and other ways of obtaining information, but it's still heavily implied that he'd have liked to go out for a stroll and help where possible.
Why couldn't he have left with a different disguise?
We see multiple options for disguises in the books:
- Invisibility Cloaks
- Polyjuice Potion
- Transfiguration (e.g. Ron's disguise as the invented Dragomir Despard during their Gringott's break-in)
Is there a reason given for why any of these, less "spottable" disguises, weren't used to allow Sirius to go out and about?