With the alabaster theme I like that you can have badges added to the sidebar (and a description), e.g.

See https://github.com/bitprophet/alabaster/blob/master/alabaster/theme.conf for all options

Implementation

The easiest way to do this would be to support the Icon Links functionality from the pydata theme:

• https://pydata-sphinx-theme.readthedocs.io/en/latest/user_guide/header-links.html#icon-links