1. bogus /wp-login.php requests, or endpoints of presumably insecure wordpress plugins. These bots are pretty dumb and do it non-stop, even if the server constantly responds with a 404
2. testing recent Apache vulnerabilities by POST-ing to something like /cgi-bin/.%2e/.%2e/.%2e/.%2e/bin/sh . Even if your web server clearly communicates that it's not Apache, the bots still insist on testing Apache vulnerabilities. They also occasionally test vulnerabilities that exist in ancient Nginx versions.
3. less common, but bots that exist to scrape something from the internet. I remember two years ago seeing a bot whose sole purpose was to document as many registered, valid domain names as possible (I found out about this since they linked a website explaining who they were in their user-agent string)
Overall, I would say the background noise of HTTP servers is tame compared to what you see for SMTP servers and, to some extent, SSH servers. I happen to also self-host e-mail; logs record failed login attempts about every second. They always pick a username like "admin" or "adm". There's also people who try using your SMTP server as a relay for spam.