This is a topic of interest to me, since I use and administer systems with all three programs and have been trying to settle on the best one to recommend. I'm just ranting long-winded here on personal experiences, but in case anyone finds this sort of thing helpful:
For a long time I've been setting up offices to use Eudora, and I very much like the portability of its data folders--as long as you're opening up the correct settings file, the folder can be anywhere. Sadly, I've been having more and more problems with it, particularly on office systems with only a few users who want to store their mail folders on a fileserver for backup purposes (yes, I know a centralized mail server would be nice, but that's not always a realistic option for a number of reasons).
My beefs with Eudora are that it tends to bog down severely on folders with over a few thousand messages, particularly over a network, and probably more importantly it is frighteningly easy to corrupt settings; I've seen three times in recent weeks on both Mac and PC versions where users lost either their main settings (and backup!) or their entire address book for no particularly good reason.
At home, I use both Eudora and Mail for different accounts, and the truth is I enjoy Mail more--at least under 10.3 and 10.4 it runs more smoothly, feels MUCH more polished, and handles HTML mail and attachments better. It also works drastically better with non-English mail, which is important to me (Eudora can handle non-English text, but it has issues, particularly in subject lines). The search features also generally, if not always, work better. Eudora has its strongpoints--speed in a few cases, bulk simple-text searching and the interface is better for sorting large volumes of mail in some cases. Oh, and Mail has a very good spam filter--several times better than Eudora's, responds well to training, and impressively accurate.
Mail of course has the disadvantage that its files go in your Library folder, period. You want to back up, you back up your Library whether you want to or not. This, for office clients, is my big beef with it and not one I've figured out a good fix for. Were it not for this and relative difficulty transeferring accounts from one user to another (something Eudra excels at), I'd probably have most Mac users on it already.
Meanwhile I'm still trying to get a feel for Thunderbird; I haven't tried the Mac version thoroughly yet, but it seems to be a more stable alternative to Eudra on the PC. It actually feels a lot more like Mail (and the hated Outlook Express) than Eudora (particuarly the funky Windows Eudora layout), but that can be a plus or minus depending on your taste. It's been pretty stable in office environments as of version 1, and I haven't seen any security issues with it, either, though a few oddball bugs have popped up here and there. The free and cross platform part is nice, too, and it can technically store its data files on a user-specificed volume (including a server), though the process is a little arcane.
In all, I think for the average Mac user you're probably going to be happy with Mail--I am, as are most people I know. If you're looking for more cross-platform goodness, I don't see any reason not to try Thunderbird, but if you don't have a particular reason to "need" it, I'd stick to Mail for simplicity and the good spam filter.
