What could be the reason for chain skipping My chain is skipping only on the smallest sprocket, and it skips with every crank turn. The chain is not stretched. Also, chainrings are almost new. Could this freewheel be the reason, it doesn't seem worn out to me (although it's old):
The reason was the sprocket after all. Changing the freewheel resolved the issue. I had some problem finding a service which has the wrench for that old Sachs/Maillard freewheels. Replaced it with standard Shimano 6-speed, and everything works smoothly now!

Thanks all for help.