Google your companies name. Considering those guys index a good chunk of the web you can, using your name as a keyword in the search would give you potential possibilities.
The question basically boils down to this. names are delegated. At the top is root, then .com. So you'd have to go to all the ones you cared about like the .com provider(verisign) .net(verisign again) and get access to their list of domains from the main databases(which must be huge and spread across multiple servers I imagine) and do a massive text based search.
1) Good luck with that. I doubt they'd help you even if you paid them loads of money
2) That search is going to be awhile and only grab certain aspects. so if your trademark is HAPPYFUNDAYOMG then searching for that wont show mispellings like HAPYFUNDAYOMG.
That's why I say just use Google.