What useful items you save or collect for re-use? Years ago folks saved  string and nails, horseshoes etc.

I save twist ties from bread wrappers, I find them useful for other things. When my kids were little, I saved lids from catsup, mustard, syrup bottles, toothpaste caps, dish soap tops, etc. With so many people using those, some were bound to get broken and I always liked to have a replacement cap.

I have a habit of keeping twist ties and rubber bands ! Sometimes they come in handy for other things.

Plastic grocery bags use them for , disposal of meat trays, hold corn meal for fish or flour for chicken and banana peels. Small waste basket liners and anything you dont want to be  open in your trash for a couple of days.

I'm with the majority on this. Elastic bands.

Like others the plastic grocery bags. Not only to line trash cans with, but I put any garbage that will smell (ripe produce, meat scraps, shrimp tails, etc.) in them and then put it in the freezer until garbage pick up day.

Pretty much what has already been mentioned.  Plus the lids from the small cans of Fancy Feast Cast Food.  They are great for plant labels - the pull tabs will usually slip over a branch or stake, and the white surface with a black permanent marker show up well.

Myself? A few of the items listed by others, plus an occasional smooth white cardboard (for artwork), and pieces of good wood.
Also I might save good giftwrap or ribbon.

I take empty Kleenex boxes and put my empty plastic bags in them. That keeps them organized and out of my way! I also collect rubber bands from the daily newspaper so my grey kitty has something to play with!