This is a Canonical Question about DNS/Hostnames resolution to IPs/Ports

Example 1

I'm running a web server on port 80 and another on port 87. I would like to use DNS so that www.example.com goes to port 87. How can I accomplish this using DNS only?

Example 2

I'm running a service on my server on a non-standard port. How can I get clients to connect to this non-standard port automatically? Can I use DNS? Is there some application specific support where DNS could indicate the IP and Port?

Example 3

Do some application protocols specifically support hostname awareness, and allow special actions to be taken based on this information? Are there other questions on Server Fault that cover some of these?

_{Commandeering: This question was originally asking about running IIS and Apache on the same server, but the same concepts can be applied to any server software receiving connections from clients. The Answers below describe the technical problems and solutions of using DNS and application protocol support to assign a port number for a client to connect.}