You can use this Domain Name Availability Checker to see if your choice of domain is available.

Leave out the ‘WWW’ and just type the second part of the domain. The checker automatically checks for dot.com, dot.eu and dot.ie so there’s no need to type the last part of the domain either. Eg to test for domains: mydomain.com, mydomain.eu  and mydomain.ie simply type ‘mydomain‘ and you will see the results right away.

You can then either contact RainRain to register the domain(s) or (in the case of dot.com or dot.eu) you can register them directly yourself via the “Godaddy” links.