Sometimes what's best for one customer will not be the best for another customer so it's hard to say that one or the other is best. As a customer myself I see and have experienced all 3 types of customer service and support.
Here's how I think about each one:
I like live chat because many times you can get an issue resolved faster than the other two options. If I have something that is going wrong with my service then I would want the fastest solution.
This is what I use for my support needs most often because I usually don't have any kind of critical support need and I prefer this method unless I have a critical need because I can submit a ticket and then go and use my valuable time doing something else.
I really don't like phone support and that's due to a couple of reasons.
One reason is that in the past when I have used phone support after my need is taken care of I get the whole sales pitch about other products or services. I didn't call to buy something but to resolve something so I don't like wasting time listening to their sales pitch. Godaddy and 2 EIG hosting companies burn me out on phone support because of their sales pitches.
Another reason is that most we hosts have phone support only during certain hours and it never fails that when I could use phone support it is "Offline". It doesn't do me much good if they don't have 24 hour support.
This is my outlook on the types of service and support from a customer point of view.
