Can a class A fire be extinguished with a class BC fire extinguisher?
[Nations vary, so these are US fire extinguisher type codes - A = wood and paper, B = grease and oil, C = Electrical]

It's always acceptable, it's just not the best choice - but if it's the only one you have and you can safely apply it, worth a shot - then if still safe, then go find a bucket and throw water at it (if its not got grease or electrical issues). Better yet, if you know the difference and can explain the difference to others in the house, pick up a nice type A (or AC these days - weird but available) water extinguisher and set it beside the BC extinguisher if the location is non-freezing - or pick up a bigger ABC powder unit for freezing locations or users that may just grab whatever there is without paying attention to the type of fire.

Generally "BC only" extinguishers are too small to do much good for an A fire - one difference in ABC dry powder types is that they are bigger, and have an A rating that's about a tenth of their BC rating. [Edit: they evidently do use a different dry chemical type, as well] Even a CO2 BC will knock back an A fire, but it will come right back when the CO2 runs out and oxygen returns to reignite the embers - still, it will have been knocked back.

ABC chemical and BC chemical

Looking at my current ABC dry chemical unit, it's rated 3 A, 40 BC (the number is larger for some arcane "size of fire that can be extinguished" system) so that one is not even a tenth of the BC rating. Thus, a small "kitchen" extinguisher that's 5 or 10 BC won't even rate a 1 A with that chemistry.

Remember in no case should you stick around to fight the fire if doing so is not clearly still safe - fire extinguishers are good for putting out small fires before they become big ones - big ones you leave and let the fire department put it out, remembering that being alive is worth more than all the stuff that's burning.