There are certainly a plethora of different collets; 5Cs and R8s being the most common for manual machines. I think my Ellis dividing head takes 3Cs.
Most NC and quick change are typically ER collets. Brown & Sharpe have or had their own collets. Jacobs used to sell a collet chuck for lathes that took collets that were made up of steel plates arranged in a circular pattern connected together with Vulcanized rubber. Those chucks and collets are very handy and typically run truer than a top quality 5C collet.
Keith is the last word on this forum on machining and especially on NC equipment. I'm counting on him to correct me if I'm giving inaccurate info.
I have 5C collets in 1/32 increments from 1" ot 1/16th, and they are very
handy for standard sized stock. I lucked into a particularly accurate 3 jaw
chuck, so they are used less than on my old lathe.
Never have machined anything as small as 1/16th. That would be a real challenge,
I think.