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
Twitter website - unlike the haters, I really like the "New Twitter" interface. I use it with the following browser extensions: