Most of the plugins I see also send the newsletter, which I don't need. I need a simple, customizable plugin that helps me get user e-mails, name, and other custom fields, and store it in the database. The plugin must also have an unsubscribe option, of course.
Contact Form 7 is pretty flexible. And there's the premium plugin, Gravity Forms, which a lot of people love. Or if you don't really need to keep the subscriber info in your own database, you could always use a service like MailChimp, for which there are WordPress plugins available for subscriber sign-up.