Are there any publicly available instances of ArcGIS Server for learning? I have searched Amazon EC2 but I don't see any public instances available (unless you already have an ArcGIS Server license).
When you say instances, I am assuming you mean to hook into and actually play with the back-end as opposed to work against the actual services that ArcGIS Server dishes out?
There are some Esri AMI templates available for you to spawn your own AWS instances up to play around with.
Assuming you have an Amazon account setup, you will need to request access to the Esri AMIs from their Customer Service team. See their quick start guide for more info.
If you are on maintenance or on their EDN programme, this will allow you access to these templates. You will need to bring your own ArcGIS Server license to the instance as well.
There may well be some custom AMI templates with ArcGIS Server on, and provided by 3rd parties, and searchable on the AWS AMI catalog. However, same applies, you will still need to bring your own license.
If it is just the services you need to work against, check out their sampleservers.