So within the past few days, I have been experimenting with pencil leads connected to a computer PSU. I have managed to create small "arcs" with them that are extremely bright (around the same brightness as a 20w metal halide at least) that only last for a few seconds. Today I decided to try and upscale a bit and so I got a dead lantern battery and took 2 carbon rods out of it. I connected them to some 12v leads from a PC power supply and touched them together. I got a few bright flashes but when I pulled them apart a little bit there was no arc. Whenever I touched them together I just got a lot of smoke and nothing else. What am I doing wrong? All of the online how-to's say this is the proper way to do it. Do the ends need to be sharpened?

Higher voltage/current supplied is the main difference i think. The computer PS have good voltage regulation, while the battery may have significant voltage drop if it is old and its internal resistance is going up

Adding an inductor in series with the arc might improve arc stability (because it will provide a kick if the arc drops out), but while the computer PS does contain inductors, there are also capacitors after them that negate this effect, so i dont think it is doing anything here

Anyway, many computer PSs dont have adequate overload protection so be aware that you may blow the PS with those experiments

I'm not too worried about blowing it since I have quite a lot of them and they already were having problems when I removed them. I might try this with a battery charger since I read that those will give better results.

In bad PSUs the capacitors are often EOL, so this means more HF ripple on the output. I wonder if that helps the arc ?

Chargers have higher OCV but also more limited current output

If the caps are old add a few large ones on the output or even add a lead acid battery like an old deep cycle battery, it will smooth the output and have no problem dealing with the increased current needed during the starting of the arc...

The main problem is, the 12V OCV is just too low to form a stable arc.
For an arc you should count at least 15V cathode fall, only the rest is on the anode column (so the main arc). That means he 12V is just not able to make any stable arc at all. Plus you need at least half of the arc voltage as ballast resistor (the carbon resistance is already part of that) drop.
With the pencil leads you have created most likely an incandescent lamp (heated by just the passing current), not an arc (heated by a discharge).

For an arc lamp you need really way higher voltage:
DC arc welders operate the arc at about 20..30V, the OCV is about 40..50V, at very low curents (for ignition) it uses to be 80..90V (or they use really high value inductor to generate that as an inductive kick - mainly the old ones).

Similar voltages are found with the arc film projector lamps (both HID, as well as carbon arc types; the HID use 10's kV ignitor on top of that).

So for such arc lamp experiments you would need at least 45V or so (for small power experiments a bunch of cheap 9V block batteries is just fine - it provides as well the ballasting resistance).