The wi-fi service is usually available on all Class 168 'Clubman' style trains and on the fancy 'Silver' trains that have electrically operated doors. If your train has tables, it should have wi-fi, unless you happen to catch the old 'blue & grey' train (with slam doors) that does one trip from Banbury to Marylebone each weekday morning, with one return trip in the evening.
The service is obtained by registering for the service the first time you log on, and then by using your email address on subsequent visits.
The wi-fi doesn't always work for everyone, usually for technical reasons.
You should also be aware that the wi-fi system on board the trains uses the mobile phone network. This means that there is not an unlimited amount of bandwidth (usage is slowed down after 20mb of data) and if the train is passing through a mobile-data Bermuda Triangle, then the data connection will drop off.
The service is provided by a company called icomera, which is based in Göteborg in Sweden. They have staff in the UK, but the company's customer Help Desk is located in Scandinavia. They are a jolly helpful and friendly bunch of people.
What support options are available when the wi-fi doesn't work for you?
Telephone: A geographical London based Help Line number was announced in September 2014. It is 020 8028 0375. [Edit Sep 2014]
Email: Provided you have 3G access (presumably the on-board wi-fi is not working), you can email firstname.lastname@example.org You should include basic information about the nature of the problem and what train you are on. It is really helpful if you can include the carriage number you are in - this isn't always obvious, but is usually above the door to the drivers cab if you are in one of the the end carriages of a Clubman 168 train.
Sending an email will automatically generate a service ticket, with an auto response email. My experience is that these tickets do usually take a few minutes to feed through the system before they come to the attention of an icomera Help Desk agent.
Internet: There is also a support website - Once you have logged an email ticket, you can respond and log further tickets via the support system. I like this because I have a record although, as with the emails, it can take a short while before your problem is picked up.
The web site also offers a Live Chat facility - of course you will need your own means of accessing the internet to use this. Also, if you lose your signal, the conversation may break off. With Live Chat, you can have the system email you a copy of your conversation.
The icomera Help Desk service is supposed to be available from 0600 to 2200 every day. Occasionally response times are not immediate, but please remember that the agent(s) on duty are likely to be multi-tasking. Very occasionally response times suggest that no one is there - When that happens, I tend to get grumpy.
Support is not currently available from icomera via Twitter although the lovely Chiltern Tweeters will occasionally log a fault - I find it much more efficient to do it myself.