Please keep in mind that the information provided here is general in nature and is not intended to take the place of legal advice. Further, trademark laws and application processes are country specific. The situations described use US law as a foundation.
If your record label is a registered business, then your label name is your "trade name". Trade names are not officially protected under copyright law, so technically speaking, someone could start a record label using the same name. That might sound a little scary, but most record labels will not meet the requirements for qualifying for a trademark. A trademark is a branding of a service or product that you can prove to be unique from what other companies are offering. There is very little room to do that as a record label.
Just because you probably won't be able to trademark your record label name doesn't mean that there aren't things you can do to keep your label branding strong. Get an internet domain name that matches (as closely as possible) your label name. Have a label logo and a distinct set of catalogue numbers. If you are an indie label, always promote your label along with your new releases. Make building your label identity a priority in case someone else does come along and think up the same name. The more established you are, the less likely someone is to run with the idea of ripping off your name. It may sound counter-intuitive - after all, your established label name may open some doors - but the jig would be up almost instantly. Anyone serious about establishing their own business will realize that they are much better off building their own label brand.