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I'm pretty sure you can check whether a domain is taken through any number of domain registration sites, but the one I've used for years is networksolutions.com. The first thing you see when you go to the home page is a "Search For a Domain Name" box. Just type in a URL and you'll know if it's taken under any number of domain extensions (.com, .net, .co, .org, .xxx, whatever).
I have heard that some domain hosts will register a domain that a user was inquiring about and then offer to sell it to the user, which strikes me as pretty sleazy. That's never happened with Network Solutions, and I've checked hundreds of domain names over the past decade or so.