I am working with high voltage arcs a lot but I have no way to measure the voltage. What is a good way to measure very high voltages (up to tens of thousands)?
One way is to use a high-voltage probe. For example if you check the B&K PR 28A Datasheet you'll see it's rated for 1kV to 40kV DC and 1kV to 28kV RMS AC. As a ballpark price they are currently $US72 at Digikey. They include a 1000:1 divider to bring the voltage down to levels that normal test equipment can deal with.
I have seen DIY solutions in the past, but of course you'd have to be very careful with the materials used and the construction so I wouldn't really recommend it. Especially if you don't have to knowledge / equipment to test insulation breakdown.
Very high voltages create strong electric fields. I wouldn't recommend physical contact with the high voltage system (I remember one high voltage test that embedded a resistor into a brick wall.) You could try an electrostatic voltmeter. These are based on the principle that like charges repel. The simplest form of this is the gold leaf electroscope. The TREK523 (a modern digital version) measures up to 20KV with a resolution of 10V. Reference below.
Sometimes you can't use a probe - if you are designing a high voltage supply - the circuit will need feedback of the output voltage and you'll need stuff like this: -
This is a picture of some Ohmcraft high-voltage potential dividers. I've specifically chosen resistor dividers because of the reasons written on the right of the picture and, the capacitance is balanced so that high voltage, high frequency AC signals will "divide down" at the same ratio as DC voltages.
If you just need to probe stuff then use a probe and take great care!!
I might think out of the box here, because I don't know if you are designing the system yourself, but if you need to measure a high voltage the best thing is to bring it down to a lower voltage more easily to measure. This only if the system allows of course and if you are designing it.
If you have nothing to do with the design or are unable to implement a lower voltage modular part in it (safety, reliability, name it..) than I would suggest a high voltage probe like the rest indicated. http://www.tek.com/high-voltage-probe-single-ended Here is a nice one that goes up to 40kV.
Watch out with the probe's capacitance. This can negatively affect your reading and can give the idea that the system is not operating as desired when it actually does. I am experiencing that capacitive loads are more and more an issue in today's technology.
Hope this helps to create an idea on how to proceed.
A cheap safe way is to make a capacitive voltage transformer. with coaxial output to scope terminated in some resistance due to wire inductance.
The choice of cap depends on expected voltage and may be created with extra safety margin with a string in series.
e.g. 10x 100 pF 3kV in series is 10 pF in a series string may be good up to 30kV and give 30V pulse output