What do you mean by "claimed a trademark"? (And what country are you in? I'm guessing USA.)
You can register a trademark - that's one thing.
You can in some cases acquire a trademark by virtue of use over a period of time. For instance, a shop might not register a logo or name when operating as a single store or even as two or three, but if someone comes along and tries to use a similar name or logo, the shop may take legal action - of course they might not win.
As rule of thumb, if you see the (R) symbol - that means the trademark has been registered, but if you see the (TM) symbol it has not, but there are bound to be many instances otherwise (because people who should know better, don't). Also, when writing about someone else's good or services, a registered trademark may be accompanied by (TM) rather than (R) since the (TM) symbol is merely acknowledging the use of someone else's mark within the article.