It turned out that Google Chrome actually didn't support this until early 2015, see Google Code issue. With newer versions it works great, for example excluding everything that contains banners:
It's possible -- at least in Chrome 58 Dev. You just need to wrap your regex with forward-slashes: /my-regex-string/
For example, this is one I'm currently using: /^(.(?!fallback font))+$/
It successfully filters out any messages that contain the substring "fallback font".
EDIT
Something else to note is that if you want to use the ^ (caret) symbol to search from the start of the log message, you have to first match the "fileName.js?someUrlParam:lineNumber " part of the string.
That is to say, the regex is matching against not just the log message, but also the stack-entry for the line which made the log.
So this is the regex I use to match all log messages where the actual message starts with "Dog":
The negative or exclusion case is much easier to write and think about when using the DevTool's native syntax. To provide the exclusion logic you need, simply use this:
-/app/ -/some\sother\sregex/
The "-" prior to the regex makes the result negative.