This was not the really hard tiles, almost more like a linoleum type.  A very old puppet, so I guess it was preserved well.

I have seen the flexible linoleum type vinyl used. The old linoleum tiles we used for printing blocks in art classes and they did crack if you bent them. The newer vinyl seems ok , and can be sewn into the mouth. It gives a flexible mouth. I prefer something a little more solid though, like card.

PuppetBakery have just seen your post regarding the plastic can holders. I personally cut them up using strips of it as the hinge for the mouth piece. I've always done a top piece and bottom piece hinged by the plastic. Does this make sense? Experiment.. It's very sturdy stuff.

Desipio - yes, you make sense. I just got a hold of some of those nifty rings and plan on abusing them thoroughly over the weekend. Thanks for the tip!

I go to lowes and get their free samples to make some of my mouths with.

Here's how I do it: https://puppetbuildingworld.com/how-to-make-a-puppet-mouth/
Here's a video tutorial I did: