Show just the current branch in Git

$ git rev-parse --abbrev-ref HEAD
master

This should work with Git 1.6.3 or newer.

With Git 2.22 (Q2 2019), you will have a simpler approach: git branch --show-current.

See commit 0ecb1fc (25 Oct 2018) by Daniels Umanovskis (umanovskis).
^{(Merged by Junio C Hamano -- gitster -- in commit 3710f60, 07 Mar 2019)}

branch: introduce --show-current display option

When called with --show-current, git branch will print the current branch name and terminate.
Only the actual name gets printed, without refs/heads.
In detached HEAD state, nothing is output.

Intended both for scripting and interactive/informative use.
Unlike git branch --list, no filtering is needed to just get the branch name.

See the original discussion on the Git mailing list in Oct. 2018, and the actual patch.

---

Warning: as mentioned in the comments by Olivier:

This does not work in every situation!
When you are for instance in a submodule, it does not work.
'git symbolic-ref --short HEAD' always works.

In Git 1.8.1 you can use the git symbolic-ref command with the "--short" option:

$ git symbolic-ref HEAD
refs/heads/develop
$ git symbolic-ref --short HEAD
develop