After seeing superb entries from @scanlime and Ben Krasnow, I was nerd sniped into entering the Flashing Light Prize competition.

This is the result of a long series of fails. My first thought was to wonder whether the traveling arc of a Jacob's Ladder had enough oomph to light an incandescent bulb. Because the arc strikes and quenches repeatedly, I though that would be a neat mechanism for flashing the bulb.

Now while there is a decent amount of energy in an arc, it's not going to light a bulb well because of an intrinsic impedance mismatch. The transformer provides high voltage at low current, but a bulb (at least the onces I had in my pile) requires substantial current at a lower voltage. Here a transformer is the perfect solution, and I cannibalized one from a small backlight inverter. This was pretty ideal: the secondary was high enough voltage to drive a small neon tube while the primary was designed to be driven at 12V. Reversing the usual use, I put a small bulb across the primary and soldered probe wires to the HV secondary terminals to catch the arc. Initial results were promising:

So the next step was to get a decent Jacob's Ladder going. I switched up the 3kV neon transformer I started with to a beefier 10kV. That did the trick:

One immediate problem: the arc would easily get "stuck" on the low-impedance path provided by the transformer secondary windings. Now it's possible I could have finessed this by fooling with the geometry of the rods and probe wires, but a more immediate problem was the arc was not lighting the bulb even though it was coupling directly through the transformer. There was simply not enough power being transferred to the primary to light the bulb.

With a little experimentation I had a hypothesis, but not a solution. My guess is that the bulb only lights when enough of the conduction wires are vaporizing which provides sufficient metal ions to increase the conductivity of the arc. It's hard to see in the top video, but the bulb lights brightest when the arcs turn yellow, indicating that metal is vaporizing rather than the purple-white of a normal (O₂-N₂) air arc.

So that was kind of a showstopper. Perhaps I could have arranged some ablative electrodes (carbon rods? Copper? Electrolyte wick?) but I was quickly losing the simplicity of a neat solution.

So I remembered something I had noticed when goofing around with the transformer some years back. I once experimented with using bead chain for the ladder electrodes, which lead to interesting behavior: when energized, the chain would start to gently swing, either from electrostatic attraction or momentum imparted by the arc or some combination. Anyway I could cheat and use this swinging motion to switch the lamp!

...which lead to another fail. While the chain was a perfectly good conductor at high voltages, it was intermittent at best at low lamp voltages -- I suspect where the chain links touch there is oxidation and or corrosion from the high voltage arcs. So back to the drawing board!

I replaced the ball chain with better-conducting brass rods hung on wires. These would start swinging with the Jacob's Ladder arc, although I had to add some weight and a washer to form a wider electrode so the arc would not get stuck at the top. One of the rods would intermittently touch the inside of a brass tube, forming a switch. I put a 12V battery in series with this switch to flash a small bulb. More elaborate than my initial idea, but such is the way of engineering.