GitHub Actions: how to target all branches EXCEPT master?

There is now a branches-ignore option:

on:
  push:
    branches-ignore:
      - master

https://docs.github.com/en/actions/reference/workflow-syntax-for-github-actions#onpushpull_requestbranchestags

Update: There is a newer filter described in Samy's answer that provides a more succinct way of achieving this.

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The documentation has been updated with more information now:

When you specify a branches or tags filter, the workflow only runs if at least one pattern matches. Any changes to branches or tags that don't match a defined pattern will not trigger a workflow. The order that you define patterns matters:

• A matching negative pattern after a positive match will exclude the ref again.
• A matching positive pattern after a negative match will include the ref again.

So in order to exclude master, you need to ensure that a pattern matching everything is included first:

on:
  push:
    branches:    
      - '*'         # matches every branch that doesn't contain a '/'
      - '*/*'       # matches every branch containing a single '/'
      - '**'        # matches every branch
      - '!master'   # excludes master