Online casinos were "born" back in 1994-6 (sorry,I don't know exactly),and,yes,there WAS some trouble with some of them.People were not paid if they won-"ghost" casinos etc.But today... you won't have any problem at all.
Online casinos win LOTS of money every day.They will not risk any chance of any kind of detraction.
I once had a small problem-some money of mine seemed to be evaporated in the cosmic dustbin-not a great amount-less than 20$.It wasn't my fault,neither theirs,just internet.Well,I sent them an e-mail.They responded within 2-3 hours,and still after the problem was solved (I had my money back),they contacted again to verify everything was ok with apologies etc.
So,all casinos mentioned by pals above are just ok.To add some:
www.bwin.com (my best currently)
www.willhill.com (the most famous in Europe)