Follow me at @sillygwailo. I've lost count of the services that post to my automated ephemera account @justagwailo, but I attempt to keep track.
Regarding the official Twitter apps for iPhone, iPad and desktop Mac, they work well with my brain despite considerable criticism of the last two. I want desperately to use them more but they don't have muting, and muting goes against Twitter's growth model, so why would they add it?
I love that you can select a list as your home timeline (especially now that native retweets are available in the API) and permanent or temporary muting of users and keywords/hashtags (especially useful for conference goers who suddenly live-tweet everything)
its integration with third party services (Instapaper, Bit.ly Pro, Tweet Marker) is excellent.
using iCloud to sync with Tweetbot for iPad
Hibari because it's incredibly simple, also has muting, and the developer has a very pleasant, supportive attitude (see the @hibariapp account for what I mean).
Twitter website - unlike the haters, I really like the "New Twitter" interface. I use it with the following browser extensions:
Slipstream to mute people and phrases
Embedly to preview links and multimedia inline. It's especially interesting when people use an external website like Posterous to to a "long tweet". (They seem to even actually have a business model!)