I am interested in finding a project (with source) for a WMS client for Android.
You could also use OpenLayers inside of a WebView, though it probably won't perform as well as amac's answer.
QGIS Mobile for Android supports WMS. It's a port of QGIS Desktop created during this years Google Summer of Code. Currently, it's only recommended for use on Android tablets since there is no dedicated mobile GUI yet.
For more info check the author's website: http://www.bernawebdesign.ch/byteblog/2011/08/24/gsoc-2011-final-report/
My multitouch mapping client kivyMaps runs on Android. It supports WMS, WMS-C and I am working on simple WFS support. It's aimed mainly at showcasing services with a clean interface, so it does not have a fully fledged UI or android settings screen, but you might just as well see this as a mapping framework and develop your own application around it.
It builds on top of kivy, so you can run the same code on Android, Windows, Linux and MacOSX (iOS is experimental).
This might help for basic needs but i cannot assure for further needs such as on click info window