EMI: Time domain (i.e. oscilloscope) measurements are more relevant than frequency-domain ones (i.e. spectrum) simply because spectrum analyzers (with the exception of really expensive ones) would miss transients which dominate the type of electrical noise present on power lines and ground. Measure both common mode (i.e. between Live/Neutral and ground) and differential (between Live and Neutral). Here is an article I wrote on the subject for inCompliance Magazine that deals with these things: https://incompliancemag.com/article/measurements-of-conducted-emi-in-the-manufacturing-environment/ The tools to use: a portable digital scope (I use Hantek 1200 family, but others can be just as good - just make sure that the bandwidth is at least 100MHz), and power line EMI Adapter MSN17 (https://www.onfilter.com/emi-adapter-msn17) otherwise you won't be able to make any relevant measurements and can kill your oscilloscope. Stay away from the tools that give you fuzzy relative data and no specification.
If you are interested in harmonics, you can use the same oscilloscope (make sure it has FFT - most do. Read the user's guide!) and 100:1 probe. Scope must be in battery-powered mode only! Unless harmonics are quite severe (at the factories I have seen quite a bit, but not at homes), they are unlikely to affect anything of importance in this case.
If you are interested in harmonics, you can use the same oscilloscope (make sure it has FFT - most do. Read the user's guide!) and 100:1 probe. Scope must be in battery-powered mode only! Unless harmonics are quite severe (at the factories I have seen quite a bit, but not at homes), they are unlikely to affect anything of importance in this case.