I pay all my bills electronically. Partly because the mailbox where I live is insecure. Partly because I work for the bank and the service is free, but mostly because the payments arrive at the biller in two days rather than five. I think I only have two bills come by paper now, most of the billers have the option of electronic presentment.
The bank where I work depends on the mail though. At least my department does. Most of our mail is notarized stuff, or checks.
Lots of our payments come by wire though. And the percentage of that is set to increase come March.
If we could get past the notary issue, we could just the post office goodbye. The competitor we just bought out doesn't do notary like we do. We are researching their way as maybe the better way, if we could just answer a couple security question problems.