The answer is to buy the original albums, NOT the ropey compilations.
If you're after bright and breezy Sinatra I can recommend Nice 'n' Easy, Come Fly With Me and Songs for Swinging Lovers
If you're after torch-song Sinatra, the bittersweet joys of Only the Lonely and In the Wee Small Hours are immense.
Ideally you should aim for the Mobile Fidelity (aka MoFi) pressings, which are superior to the standard CDs, but the ordinary ones are great, too - the original quality of the Capitol recordings is so excellent, it's hard to muck them up (except when you try and cram 20+ tracks onto a compilation disc...).
Anyway, you could have a punt on one of the original albums for under £5 on Amazon. Stunning value....