There is a lot of glossing over of details going on for my part, but this is something that I meant to say and it did get omitted.
A decent modern motherboard will only output a PWM signal if it detects a PWM capable fan. I would try and grab a 4 pin extension cable to make measurements easier, but you probably need a fan plugged in to see anything. A circuit to fake there being a fan connected would cost more to make than cheap fan
In that circumstance, I would totally expect to be able to see the PWM signal even on one of those cheap multimeters. He isn't talking about detailed duty cycle measurement here, just a basic "is this actually PWM" which should be pretty obvious compared to a DC fan control with just the voltage varying.
That DMM says it can do frequency and duty cycle measurement, in which case it should be able to tell you what's going on, or at least I expect I could work out what it is doing with that but I'm not sure how clear it would be to a beginner. It looks a useful bit of kit regardless.
That diy scope still looks rather toy like. Having some sort of 'scope is really nice though, nothing beats being able to actually see the waveforms coming out, specially if a circuit was designed by someone like me where I have used some pretty whacky waveforms for driving things like LEDs because I ran out of hardware timers and had to drive off a 1KHz interrupt in software, but it saved a few pence doing it that way which adds up when you sell a million units
If you want something that could be genuinely useful in the future, you sadly need to spend proper money, something like this:
https://uk.farnell.com/multicomp-pro...sps/dp/3107575
where a PC does the display and user input bits, so you are only paying for measurement electronics. at work we use similar Picoscope kit, I think we only have one "proper" scope with a display on it.