The latest trend is for Greatest Hits for an artist. Unlike rock groups today, most jazz musicians of that time period jumped from label to label so a Greatest Hits compilation probably won't mean much.

Not quite. Often, as record companies have merged and gobbled up each others rights, you end up with some companies owning parts of an artist's recordings, while some other company owns the remainder.

Example: let's say you like "old testament" Count Basie (that is, recordings before WWII). You can get the complete Decca recordings (a 3CD set for a pleasant $35) which encompases three years worth of 78's, or you can get something like Volume 4 - 1937 with live radio recordings. And, of course, after Basie got out of his evil Decca contract (record companies have been outnegotiating quality artists since the beginning) and signed with Columbia, you get different compilations like The Essential Count Basie with material from 1939-1940.

As a general rule for post-war music, I try to stay away from "best of" collections and focus on original albums. If you want pre-war music, you have to remember that it was all issued on 78's -- two three-minute cuts per disc. If they were lucky, they did the remastering from the original metal parts. If you're unlucky, they did the remastering from a record that hopefully hadn't been overplayed. Plus, back in the day, radio stations would broadcast live shows most every night, and some enterprising people thankfully taped those shows. For example, I highly recommend Benny Goodman On the Air (1937-1938), which is restored entirely from old radio bootlegs. Radio shows may or may not have better fidelity than 78's, but they often have more energy, plus they're not limited to 3 minutes per side.

One label I've been particularly happy with is JSP, a British firm which makes nice box sets and is known for the quality of its digital remastering. I have their Cab Calloway, Jell Roll Morton, and Louis Armstrong Hot Fives & Sevens. All quality work. For contrast, the Chronological Classics label has the most stuff in print from the 1930's, but they don't even attempt to remaster the music. You get all the original album popping and hiss.