More equipment dies at the bench than in the arc, and the reason is always the same: a scope's ground is bonded to mains earth, and the thing you are probing is not.
The one rule
For a high voltage input, use a differential probe. A Micsig DP10013 is the budget end and a used Tektronix P5200 the classic. Failing that, a 100x probe with a real CAT rating.
Probing the driver
Ground the probe to the driver board's ground and nowhere else. A probe ground clipped to the driver's output is a set of dead output transistors, and there is a whole thread of somebody discovering this.
Each GDT input gets its own probe. Do not share a ground clip across two points that are not the same point.
The two burdens are not the same
- The feedback burden, around 51 Ω, has one side on ground. An ordinary probe is fine, and it reads the current correctly, because the series phase lead inductor does not affect it.
- The overcurrent burden, around 5.1 Ω, has neither side on ground. Connecting both sides of an ordinary probe grounds one of them through the probe, which distorts the waveform and disables the protection while you watch. It needs a differential probe.
That distinction has cost people bridges, because the protection was off exactly while they were looking at why the protection was not working.
Looking at the bridge output
Two probes, one on each output, both ground leads on the same node, the negative bus. Subtract in the scope. That gives you the differential without a differential probe, as long as the supply is floating or earthed consistently.
Looking at a gate
The Miller step on Vge is normal and not a fault.
The current transformer, while you are in there
And a CT belongs on the lead between the bridge and the primary, never between the primary and the tank capacitor, where there are kilovolts. There is a case on record of an inch of arc onto a probe.
Making your own measurement transformer
Two or three ferrite toroids cascaded cover 100 to 200 kHz. davekni's reference build is an E80 core with a one-turn split foil screen and a foil primary, which reads 1 V per 100 A cleanly up to 2600 A. Nanocrystalline is the best material. A Pearson is the gold standard if one turns up second hand.
The arithmetic that has to be redone after every upgrade
Burden resistors dissipate. A worked failure: a 51 Ω, 2 W feedback resistor at 2.9 A peak and 17 per cent duty is 16 W, which is a fire.
Recompute the burden's power every time you raise the current limit. And the overcurrent threshold itself is V_ref / R_burden × CT ratio, with the reference typically around 4.1 V from a 2.2k and 10k divider off 5 V.
What can hurt you rather than the equipment is on its own page, and it should be read first.