The solution? Disposable email addresses. Services like Spam Motel, Sneakemail, spamgourmet and EndJunk allow you to create as many forwarding addresses as you need and turn them off if you start getting spam through those. Of those, spamgourmet seems to offer the most control, including creating email addresses that automatically shut off after a certain number of messages. I use EndJunk myself for its simplicity.
Found two more of those services today. Spamhole creates email addresses that remain active only for the specific amount of time you choose.
Mailinator doesn't require a signup process at all. Just make up any Mailinator address you want. The downside is all accounts are essentially public. Anyone can take a look if they know what ID you have chosen. But if you use a hard-to-guess ID and if it's just to pick up one or two emails, that shouldn't be a problem.