I have used several types of lights mounted on the side of a bucket and they don't seem to work out. I love the Streamlite rechargeable lights, especially the newer LED ones. I had a plastic holder that it mounted to but got rid of it. The small Streamlight hardhat lights are the best that I have used and swear by them.
I looked at the streamlight and Pelican flashlights for work before I decided on trying to get these.. the Milwaukee's were brighter and had longer runtimes for all the other models with similar sizes to these.. I also liked that it was easy to just grab a second battery to get 8 hours out of em if need be, not to mention the spot and flood feature the Milwaukee's provide..
At work, they just recently started to provide streamlight (I think) hardhat lights. I honestly don't like the model they have. It's a cheaper one, like $30 or so if I remember. It's a solid light, but it's a spot light. So you get that one annoying bright spot in the middle that your eyes always follow, and you wind up moving your head around to point the bright spot on things, instead of being able to just move your eyes like you can with a flood light.
I bought a Coast model HL8 I think, and I love it.. 3 other people bought the same after seeing how good it was. It carries 4 AA batteries in a seperate battery housing at the rear of the hat (you can wear it on a belt too, and run the extension wire they provide) for longer runtime, and it's so bright (344 lumen) that I can run it on the medium setting and it'll last all night long. We have batteries at work, so I don't care about needing to replace, and I don't have the time to wait for it to recharge like some other models. It has a focusing lens too, so I can adjust from a wide flood to a long penetrating spot light if need be - like if I'm on a digger and setting a pole at night between live power lines.
Even the smaller cheaper Coast lights are good, you just lose the runtime and are swapping batteries every 2 or 3 hours. Coast makes the headlights for LED Lenser, which is owned by Leatherman.. the only downside, is I can't use the hardhat clip Coast provides with the safari style hardhats I use. The belt openings are on the wrong side, but for $5 or $10 I found a better set on Amazon with the openings on the other side.
This is the Coast light I have (the lumen rating is wrong on the page)
HL8 Focusing 615 Lumen LED Headlamp
https://www.amazon.com/dp/B00A7RM1JE/?tag=atomicindus08-20
These are the hardhat clips I use
Petzl crochlamp HEADLAMP CLIPS E04350
https://www.amazon.com/dp/B001GCGSKW/?tag=atomicindus08-20