Do we know anything about Sirius Black's profession, or did he just live off his inherited belongings before he went to Azkaban?
He must have been free for quite a period between getting a home from his uncle and getting into Azkaban.
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Sirius's Uncle Alphard left him a lot of money towards the end of his stay at Hogwarts. This was enough for him to buy a house, and later a Firebolt, with the remaining funds still adding up to a significant percentage of the Potters' fortune.
“Where did you go?” asked Harry, staring at him.
“Your dad’s place,” said Sirius. “Your grandparents were really good about it; they sort of adopted me as a second son. Yeah, I camped out at your dad’s during the school holidays, and then when I was seventeen I got a place of my own, my Uncle Alphard had left me a decent bit of gold — he’s been wiped off here too, that’s probably why — anyway, after that I looked after myself. I was always welcome at Mr. and Mrs. Potter’s for Sunday lunch, though.” Harry Potter and the Order of the Phoenix - Chapter 6
Immediately after Hogwarts, Sirius joined Dumbledore's renegade Order of the Phoenix to fight against Voldemort, and wouldn't have had time for any major jobs during the three years before he went to Azkaban.
By the time the four friends left school, Lord Voldemort’s ascendancy was almost complete. True resistance to him was concentrated in the underground organisation called the Order of the Phoenix, which all four young men joined.
Pottermore - Remus Lupin
We don't even know if he had one at all. The only thing we know for sure is that he eventually became self-sufficient (emphasis mine):
[W]hen I was seventeen I got a place of my own, my Uncle Alphard had left me a decent bit of gold — he's been wiped off here too, that's probably why — anyway, after that I looked after myself.
Order of the Phoenix Chapter 6: "The Noble and Most Ancient House of Black"
But whether he looked after himself" with Alphard's generosity, or if he buckled down and got a job1, is something we can't confirm one way or the other.
1 Heck, for all we know he might have done both
He worked for the Order full time. Although it is possible he was living off of gold his uncle Alphard left him, it is also possible that James supported Sirius financially as he did with Lupin. He might have also saved money from a job he had when he was younger because even if he did not have a lot saved up he might have stayed at the Order HQ. The last possibility is that the Order of the Phoenix paid him, but I find that unlikely as the Order was a volunteer group.