Oscilloscope Probe Noise and Safety
5mV per div is a really small scale, unless you've got really good grounding and a low impedance ground connection (that clip-on fly lead isn't a good ground at higher frequencies) you'll see noise as the loop area in the fly lead basically acts like an AM antenna (which are just wire coils too) and it picks up all kinds of crud from the airways, hence all the noise you're seeing on the scope.

A floating scope isn't necessarily a hazard provided that there isn't a low impedance path from mains to come into contact with something or someone. It's when there's unacceptably high leakage in the power supply that you'll have a problem as it'll allow enough mains to get through that it might give someone a zap (you could try doing a current measurement from scope ground to earth - like a tap or something, I'd be weary if it reads more than a few uA). Almost every laptop and phone charger in existence is floating and will have no direct connection to mains earth so floating supplies aren't a hazard on their own, provided that they're built properly.

Battery powered or isolated stuff is safe to measure with just about any scope as there's no direct path to ground, it's when you have non-isolated stuff (and a non-floating scope) that you can run into problems, I've seen the odd mildly exploded scope because someone was measuring the output of a mains rectifier and connected scope ground to rectifier ground (which will actually be -1/2 mains voltage below ground, hence the explosions). Having said that, if you've got mains leakage then your battery/isolated item will now be floating around mains voltage so be careful.

500V is probably getting into the realm where I wouldn't trust most cheap scope probes to handle the voltage well, there are plenty of HV probes which offer an extra level of safety through higher levels of isolation. One of the best things you can do is to stand on a nice rubber mat and keep one hand in your pocket when probing the circuit, that more or less eliminates any direct current paths through your body. Current from one hand to the other is the most dangerous as the current will be passing through your heart.