There have been a couple of US court rulings (2002, 2007) allowing the re-use of thumbnails of copyrighted images on search engine websites. Thumbnails have been defined as maximum 200 pixels each side - a resolution which has little value for reproduction.

Does anyone know what the situation would be for the re-use of copyrighted museum images as thumbnails on a non-commercial site?

The sort of scenario I can think of is where the data from one database is exposed through an API to another website (possibly a portal representing multiple providers or a mashup of the original data to give a different way to navigate it). Use of the thumbnail images would give the third party site a lot more potential to create an interesting interface, but I'm not sure if this goes beyond fair use. The third party site would of course be prevented from showing larger copyrighted images (this goes beyond fair use). These could only be shown by linking through to the original site.

We're on the side of the API provider.

This exact scenario - a portal offering the data of multiple providers through a set of APIs - is what Europeana is doing, and obviously the issue of rights (and not just re thumbnails) is high on their agenda. For the purposes of Europeana's own sites, of course, things are pretty straightforward, but when it comes to allowing their reuse via the API things get more complex. In the current phase of the project (building the full-scale operational service), they aren't taking the risk that fair use will be enough cover, or at least there is a group that will be looking hard at that question. They are also considering a policy that requires that contributors promise not to change their licencing or copyright claims to be more restrictive once they have been included in the index, though I can't even guess how this will pan out. Europeana will be an aggregator and redistributor first and foremost, with a set of user-facing services built on top but fundamentally based on the assumption that what comes in will be let go of, so obviously the licensing question is vital.