Should I use md5 for my new application? MD5 is a very fast hash and its output is short and easy to store. Should I use it for my non-security-sensitive application?

I don't think so, but I can't seem to convince anyone else that this is true.
When I need to show to others that they are wrong and it is their word against mine, I look to the words of industry recognized experts and then challenge them to find an industry recognized expert that supports their belief.

In this case, Bruce Schneier has written a lot on MD5 (in Bruce style, each word links to a different blog post or article of his).

Wikipedia even has a nice article on MD5 security.

That said, you specify in your question that this is potentially for a "non-security-sensitive" application. What do you mean by that? Your comment seems to suggest that you are signing hashes of files. That to me seems security sensitive.