I'm a daily user of Twitter and find it very useful overall. It is true that it is not at all effective for direct, threaded conversations / communication – but I believe it is very effective in other ways. Such as:
— For finding answers quickly, especially on 'techie' subjects. I have often got much quicker answers to a simple how-to type question via Twitter than I would have via Google. Yesterday I was at the other end – providing a quick answer for a user asking about whether you could transfer Firefox bookmarks from one PC to another.
— For connecting with people who share interests with you. Again, this is especially true if you are into web-related subjects and techie subjects, but goes beyond that as well. Two examples – I have discovered more interesting new iPhone apps / developers and even artists via Twitter than any other medium, by a long, long way. I come across new devs, or chat to interesting devs every day. 2nd example = indulding my love of sports. During the NBA Playoffs, when there was a game on just about every night, I would always fire up a Twitter client at some point during the games and see what folks were saying about things, and throw in my two cents.
— As a news source. Twitter is – to my mind – probably the best medium for true citizen journalism. Any number of major news items have been covered first on Twitter, by real people at the scene, before any major news outlet had the stories. This is true for the earthquake in China, the Mumbai terror attacks, the plane that went down in the Hudson, and many other stories.
For me the trade-off for Twitter's weakness as a threaded, direct conversational tool is that it offers you a very low-commitment relationship. I don't look at my Twitter clients – on the desktop or iPhone – often – just a few times per day. I dive in when I feel like – and take pot luck on what's being discussed or who is online when I choose to dive in – and that's fine. It's not a medium, for me, where it needs constant attention – and if something is directed towards me, via an @mention or a direct message, I'll always see it and be able to respond.

I understand that it is like a status indicator and allows for people to follow real-time updates, but fail to see its feasibility as communication. For example, the timeline views are an amorphous blob of random replies without any context of the conversations. In order to see what each reply is in reference to, you have to navigate a hierarchy, attempt to match the reply timeframe to the corresponding status update timeframe as to guess which status the mention was actually replying to, then come back and start again. It seems like a poor communications medium and a giant waste of time to try and follow the conversations at all.

Unfortunately sometimes people sometimes do not just reply to somebody, but get into entire conversations of reply-reply-reply. That's where i also lose control, and i just skip those messages.

If you're looking for a client with better conversation support, go check out Twitterrific. It can view a tweet conversation by looking up all the replys and showing them on one page.