I can't NOT have a bathmat! Particually in Melbourne in winter! You're all nice and toasty warm after a shower, why ruin it by stepping straight out onto cold tiles and instantly getting frozen feet?! LOL.
I have always used them, and never had a problem with mould at all. Although there have always been heating vents in our bathrooms, so perhaps that helps to keep the mould at bay? they are also not left on the floor, but hung up over the side of the shower recess, or bath. I used to use ones that were the same as a towel, but I now use the thicker cotton ones, and are perfect on a winters' morning!
This is my latest purchase:
...and hanging up on the shower recess, reflected in the mirror (which is actually close to the heating vent in the ceiling!):
They get washed with the towels once a week...but do take longer to dry as they are so thick!
I don't use a mat to stand on in the shower itself, as the tiles are not slippery.