How to test op amp stability?
This is normally (all situations that I've witnessed or read about) a stable configuration. The op-amp would be stable with direct feedback so the question is what does the MOSFET add in terms of gain or phase that might make the circuit unstable.

Well, in a source follower configuration the gain of the MOSFET is a little less than 1 so on that score the circuit is still going to be stable. As regards phase shift from gate to source there will be a little but given that the gain has probably reduced about a dB and that the MOSFET is going to be much faster (as a singular device) compared to the op-amp, I really don't think you would have any problems.

It's the sort of circuit that I wouldn't hesitate to build and not expect a decent result. However, in the circuit you have shown I wouldn't use the op-07. It cannot adequately drive its output down to 0V (read the data sheet) and this could mean that the FET is not turned off properly and that control is lost when trying to control small currents.

The same is true (if not more so) when looking at the input voltage range that device is capable of. If you are trying to control 1A thru the 1R sense resistor, the voltage at the OP-07 input is expected to be 1V and this is right on the limit of what the input range can be expected to handle (again read the data sheet).

So my conclusion is don't use an OP-07 or power it with a small negative supply instead of having its neg supply terminal at 0V.