It is difficult to say what signal levels might be a problem. Normally the problem is common-mode noise in the system since the test equipment will be earthed. For broad band noise(but not down to DC) I use a small RF transformer stuck onto a piece of pcb and feeding coax into the scope or spectrum analyser. The RF transformer gives the single-ended scope/analyser a differential input. However this type of assembly might be a bit ‘technical’ for a mechanical engineer.
You should start off using a 10:1 scope probe, but throw away the earth clip, dismantyle the probe and connect to the inner grounding ring. If that is still too noisy then go for a straight coax lead soldered to the pcb, with say 470R in series with the signal wire at the signal source end. This again makes a 10:1 (-ish) probe but with better common-mode performance.
After that you are going to have to buy a differential probe, but now you are probably talking about £1000+
This is not a trivial measurement problem.