I finally got round to digging a trench alongside a clump of bamboo in order to put a panel in to prevent it spreading. I came across a length of rhizome with a couple of shoots on it on top and thin white roots growing from under it. I have cut off and removed the rhizome. What I'm wondering is, if bits of the roots are left behind, will they grow into new bamboo or is it just bits of rhizome they regrow from?

From what I recall when removing ours, you would need a piece of the rhizome. Depending on the type of bamboo you have, you will need a barrier that goes down somewhere between 6 and 8 feet. Also, some varieties will go over the barrier.

If yours is a running bamboo it can take years to eradicate it. I had one that was running under a block wall with a very deep foundation.

The clump I have has been there for just a few years but hasn't increased particularly quickly and hasn't (visibly) spread away from its base. The rhizome I came across wasn't too far below the surface so that wouldn't be the one that grows down deep, although since I'm doing it I'll take the barrier as far down as I can, and leave a bit above ground.

I got rid of a bamboo I didn't like by spraying the rhizomes with glyphosate concentrate. I believe you could keep it from spreading in places you don't want it by doing the same.
I also got rid of a Musa basjoo shoot with the same method. Didn't hurt the rest of the banana at all.

Glyphosate does travel to and through the root system, so could kill or damage the entire plant. It is a good way to kill unwanted plants or stumps, but it has to transfer through to the root, and means pouring on FRESH cuts. I used to follow the tree crew with a spray bottle when they were removing trees, and cutting an X in the stump gives the chemical more access. I was a licensed applicator for 15 years, so I've had some experience. I still use RoundUp where appropriate , but, like any cure that has been over-used, some of the targets have developed immunity. Add to that, as all chemicals, it has a shelf life, meaning, like old drugs it looses potency after time. Unfortunately, there is no "use by" or "sell by" dates on garden chemicals .Don't get me wrong. I love the stuff, and use it myself, but you need to know what you're doing.

As for my banana it only killed the one shoot I put it on. The others right next to it were fine. Having said that, I wouldn't guarantee the same results for other plants.
I agree -- caution advised.