Yahoo, MSN, Gmail AOL who is you Email Provider. Gmail should be the Main one in your Incredible. But if you have one of those go to them on your PC and Log in then go to your Messages and Delete them..
I'm sorry to learn of the email issues you are experiencing while trying to delete messages. What's your email provider? I recommend deleting your email account from your phone and add it again. If this is the only email account on your phone you will need to do factory reset http://goo.gl/vmBVV to remove email account. Keep us posted.
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I think I know what techweld means. I use a gmail account. When I get an email, sometimes the notification icon is the gmail icon, sometimes it's the "mail" icon (that's with the phone, messages, camera icon on the Home screen). When I go to my mail using that icon it shows my inbox full of messages. When I go to my email using the gmail icon, there are no messages.....this after I deleted all messages in gmail from my desktop.
Even hitting refresh didn't delete them.