I guess it would vary, depending on the specifics of the situation ... maybe $100 or less for short-term, and I would think somewhere in the $500 to $1000 range might be good for a more long-term event.
I would opt for carrying smaller bills; heavy on $1s, $5s, and $10s, with nothing over $20 bills.
And spread the cash around amongst your carry if possible; so there is less chance of losing all of it at once.

Some coins may be useful, if you expect having to feed a machine that does not accept bills.
And, I suppose you might want to carry some old Silver coins; if the bug-out involves some sort of economic collapse scenario. [Yikes !!!]