I posted a few results on an earlier thread:
https://holographyforum.org/forum/viewt ... f=5&t=9950
With regards to resolving modes, I think it would depend on the mode spacing. It's difficult to say anything based on my limited testing, but I checked a laser diode which
seemed to have a mode spacing of more than 1nm, and the real-time curve on the Theremino software showed a double-spiked curve, with the two spikes going up and down (alternately). It kind of looked like it was see-sawing back and forth. I took this for two modes which kept switching back and forth, but I can't say for sure. The HeNe curve looked much different, and didn't jump around like that.
Thanks for the links, there has been some forum discussion in the past regarding them. The thing I like about the diffraction-grating spectrometer is that it's simple, and can be used for other purposes, like checking white LEDs, etc. Based on the discussions I've seen, they appear to be limited to a resolution of about 1nm. For greater resolution, other methods (like in the links you provided) would be needed. But, I have to say that the diffraction grating/webcam arrangement does a very good job for many purposes, and the Theremino software is free and works well.