I have a 50 amp sub panel in my basement with 6 circuits in it, which is not enough. How many circuits can I run with a 50 amp service?
You could theoretically run an infinite number of circuits with a 50A service, provided that no more than 50A is ever drawn at any one time.
More practically, if your panel only contains full-width single-pole or double-pole breakers, you may be able to swap out one or more existing breakers in favor of half-width breakers. That would double your possibilities to a total of 12 branch circuits.
Another alternative would be to install a sub-panel, which is a standalone breaker panel fed by a double-pole breaker in the main panel. You do lose two slots in the main panel, but in return you get as many slots as the sub-panel supports - whether full-width or half-width breakers are used.
For example (probably pretty extreme, for a 50A service), you could swap out four full-width breakers in the main panel for half-width breakers and gain four new branch circuits. You could then swap out the remaining two breakers for a double-pole breaker and use it to feed a six-slot subpanel. You'd migrate those last two breakers to the subpanel, leaving another four new slots available, and then you could populate those four slots with eight half-width breakers to gain yet another eight new branch circuits - total twelve new branch circuits.
Don't go overboard, though, be sensible about it. Adding breakers doesn't increase the total amount of current you can draw. All it does is open up options as to where you can draw that current.