To get really pedantic: you are allowed to put a device that normally requires a safety ground into a non-grounded ('old style') outlet but then that circuit should have ground fault protection installed and conversely, if you have a circuit (such as for instance an oven) that does not have a ground fault protector you must use the ground.

This covers a very large percentage of all normally occurring situations, though of course there will always be tricky ones due to rare combinations of devices and installation details.

If you can play it safe: put ground fault protectors on all of your circuits except where leakage current is too high as a result of the consumers design and limit those to one consumer per circuit (dedicated circuit).

This goes double for 'wet' spaces such as bathrooms, kitchens and so on, ground really is mandatory there from a safety perspective even if the code says it isn't, so if you still have ungrounded sockets in bathroom or kitchen add a ground if there is any reasonable way of doing so.

In bathrooms the ground point can frequently be found behind the bathroom mirror over the sink. And don't be tempted to use the waterline, those are frequently made of plastic nowadays and no longer count as safety ground.