Not everybody uses Twitter in the same way. Some people seem to bleat on about every mundane aspect of their existence. Others like to post links and snippets of info they think will be useful to others. Some people seem content with reading and following others, and some people just like to exchange banter with their mates. Some users are apparently 'social media gurus', and therefore should probably be avoided.
Whatever the reason you decide to use Twitter, a great piece of client software should be able to cope with all these usage scenarios. TweetDeck is the software with the greatest number of features crammed into it. It's been designed well enough to offer all this functionality without being too difficult to use, but it is a bit of a screen hog – to get the best out of it you'll need to dedicate a desktop to it, which might be a bit much for any but the most dedicated Tweeters.
We also have a funny taste in our mouth over having to register to get some of the functionality, as well as the general un-Linuxyness of it all. For that reason we have to give the overall thumbs-up to Choqok. It isn't perfect, and could do with extending its abilities, particularly in some of the areas of general helpfulness, but it's deeply wonderful and scores just as highly as TweetDeck.
Spaz also deserves a mention. Since it's driven by CSS styles it is themable, so if none of the other clients suit you, a few hours' work could bring forth the client of your dreams. Mixero has most of the features of TweetDeck, but is a lot more compact and seemingly more responsive too.
As all the clients are free to use though, you can test drive the ones you like the sound of – don't run them all at once though – that way leads to madness!