With enough experience with ring testing similar transformers, and if in doubt, by applying a shorting turn threaded through the core to set a 'known bad' state for comparison, you can definitely identify most bad ones. However proving that one is O.K is a lot harder.
For line frequency transformers, the best option (if you don't intend to attempt to rewind it) is to remove it from the equipment, and smell it! If it smells burnt it probably is and you should never trust it again.
If it doesn't smell burnt place it on a loose concrete slab outside (or other non-flammable sacrificial surface), in a location safe from pets and children, bypass the thermal fuse if its blown and hook it up to power with an in-line fuse, and wait and see if it cooks itself with no load on the secondary. If it runs for an hour without overheating, the original fault must have been due to an overload on the secondary. If it catches on fire and blows the fuse it must have been bad. Don't try to do the test inside unless you've got a good fume cupboard and some way of containing the expected transformer fire. Testing it in-situ is not advisable as the heat and smoke may cause extensive damage.
If it turns out to be O.K., don't forget to replace the thermal fuse and trace and rectify the cause of the overload before refitting it.