When I hear "use an isolation transformer", I wonder if I need to isolate only the device under test, the scope or both. What should I do?
Keep in mind that Isolation transformers aren't cheap around these parts so I would prefer to get one with a lower VA rating just for the oscilloscope.

Standard safety practices dictate to always isolate the device under test, the grounding on the instruments should stay unchanged.
One of the reason is, to prevent the instrument "ground" to become live vs normal ground, so pose a risk of an electric shock.
The thing is, as a technician working on live circuit you are expected to anticipate the voltages present within the tested device to be dangerous, but not on a seemingly unconnected instrument lead or even its case.

There are instruments that are explicitely rated to have fully isolated inputs, with those you may work on a device without an insulation transformer, but obviously only within the instrument CAT rating. Multimeters rated like that are common place, but more complex instruments like oscilloscopes or generators aren't, so such rated ones use to be insanely expensive (if even rated beyond SELV at all). And use only proprietary (so very expensive) probes/cables, as the common BNC connector design can not be made safe (preventing you from touching even the "ground" side, as that could become live with some measurement arrangements), without breaking its compatibility with other BNC connectors.

There are two ways to work with live equipment which are generally considered safe. (1) using an isolating transformer with the equipment you work on, and (2) using a high-voltage differential probe in the oscilloscope.

Using both practices *does not* make the equipment you work safe to touch in any way, but at least keeps the scope reliably grounded.

The thing to remember is even if you use an isolating transformer, once you connect a grounded scope to the circuit you work on, you create a current path to the ground and touching the circuit (but not the scope) becomes dangerous again. On the positive side, isolating transformer also significantly limits  possible short circuit currents, so you have much less bang before your face if something suddenly goes wrong  

Also putting a sensitive RCD and low-amp fuse/circuit breaker/500W incandescent in the circuit you work with is a good idea to survive too.

Using a scope disconnected from the ground or a battery powered scope is generally frowned upon, as it makes scope chassis live. Yes I can confess doing this sometimes, and being still alive, but this is definitely not recommended for novices.

 

The differential probe would be even more expensive than an isolation transformer. And I know that back a while ago way before I was born, people would cut the ground pin off of their oscilloscope so that they could work on CRT televisions and radios that had a live chassis.

So beware of Darwin who is ready to prune the gene pool haha

The differential probe would be even more expensive than an isolation transformer.

Yes, the transformer is indeed the cheapest option, that is why it was first on the "recommended" options...
If a transformer costs about $100, a basic differentiual probe would be around $1k and an oscilloscope with individually insulated inputs in the $10k..$100k ballpark range.

And as RRK stated, many of us are guilty of using a battery oscilloscope, or cutting its PE lead, it is really highly not recommended way. Quite regularly it ends up by blown instrument with a repair price tag in the $1k range. Don't have to ask me how I know, I guess...

@Medved
I don't know how you are inventing these prices, today quite reputable differential probes made by Micsig cost just around $200. Still a bit steep for a hobby use, but we are talking about something in 100MHz 1kV (and better) range, much overkill just for playing with lamps, even on HF gear.

As for isolating transformer, surplus devices may be a way to go, also no one prohibits running multiple transformers in parallel. Running ten 'hotel shaver' transformers in parallel will give ~150W power almost for free.

On this picture, I  am running a pair of 60W toroidal isolating transformers to probe the ballast of ~150W induction lamp. Sorry for a bit of mess )) These transformers were a bit overpriced for a purchase, but still well below $100 for both new.

The thing is the most important parameter here is not the frequency range alone, but mainly the probe CMRR: If you want to measure currents via a shunt, we are talking about measuring about 1V with 100..200V common mode swing on it. So in order to get the result within 10% accuracy (which is very poor, it even nearly does not allow to guess any shape related parameters), you need the CMRR to be better than 60dB. And that is beyond what most probes allow (practical experience). Such use is just not what typical differential probes are meant for. The ones with good enough CMRR to really "replace" the isolation transformer (so CMRR of 80dB and more) become quite expens8ve regardless of the frequency range.

Plus there is another pitfall of using differential probe instead of insulated DUT or the measurement channel: With electronic ballasts you are working with a rather sensitive circuits. So connecting something with few 10's pF and even 10MOhm impedance towards a potential that is swinging 150..300V against the local "ground" of the circuit, you are injecting quite severe disturbance to it, good luck it survives, not speaking about operating normally. With a proper insulated inlut (insulated input scope, isolation amplifier, isolating the DUT) you connect the signal ground of the instrument input with the signal ground of the DUT, all impedances are referred to that common "signal ground", so no "foreign" distrurbances are injected into the circuit.

You may find some insulating amplifier probes, but those become quite expensive or really limited in frequency range. And often have many other limitations too (usually made as a special probes for specific task).

Often you could be better with custom made couplers (like a current transformer for measuring currents - e.g. made from cheap clamp meters like DT87 or so, accepting the loss of the true DC component) and/or using "subtract" function between two channels (actually I havd way better experience with this than with a purpose made differential probe, talking about debugging EMC issues on communication interfaces like CAN, where you get 3V differential signal on a 40V peak rf disturbance voltage, include the limited ADC resolution). Yes, you need a 4-channel scope for that, so again extra cost for a DYI.

The isolation transformer is still the cheapest way without heavily sacrificing on what you can measure, that is whe there is not much reasonable "alternatives" (parameters, universality of use and pricing) on the test equipment market.
What you may find are special things for specific use case where it is the only option (you need higher CAT insulated inputs on an oscilloscope when you need to measure around gate drivers of e.g. VFD motor inverters, when you need to display what happens around both the top and bottom switch transistors simultaneously, mainly for some nonrepetitive event). These are rather niche cases, with few pieces of the equipment sold, hence the high cost and a dedicated, single pupose targetted functionality.

Well, I beleive @HIDlad001 originally asked for a simple probe/method to mess mainly with line frequency magnetic gear. Be real.   

Contrary to that you have said, there are very few sensitive places in the electronic ballast which will be seriously disturbed just by hooking a ~40cm wire. RC timing networks at the oscillators may come to mind, but they are almost 100% local ground referenced, so no need for a differential probe. For robust transformer-driven bipolar circuits, just none at all...