In my main bathroom, where nobody actually uses the shower, I have a decorative polypropylene oval rug there which I think looks nicer than the standard bath mat.
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However, in our master, I have a standard bath mat by the tub. I'm getting ready for a mini redo soon and I know I'll need to get a new rug but I'm not sure which type I'll get.

I honestly think a polypropylene type rug would be fine as long as you get a rug pad and aren't getting it soaking wet all the time.

As far as the number of mats, I think it depends on the size of your bathroom. Mine are both pretty small and I tried to rugs in the one (by tub and sink) and it looked odd to me.

We absolutely use bath mats. Ours are just the regular white kind, easy to bleach. In the guest bath, it hangs over the edge of the tub, hidden by the shower curtain. In the master bath, it's larger, and is in front of the shower door. If it's in the wash, I miss it. I don't like stepping out onto cold floor, and no one sees it but us.
Dee

I absolutely have a mat in front of the vanity in all bathrooms- with kids glopping toothpaste and who knows what else it's nice to be able to toss it in the wash.

I have fun, textural mats- long loopy chenille and that sort. With kids, a non skid back is essential.

for in front of the tub, I just have a more towel like mat that is folded on the edge of the tub and hidden by the shower curtain when not in use.

I have washable mats in front of both shower/tubs. More comfortable than cold tile & safer with wet feet. Mine is more like a thick towel and in DH's bath it's a non-skid mat.

olychick, your bathroom is lovely.

I have non-skid rugs in front of sinks, but I would not be without a mat that is folded over the tub when not being used. In the master, we fold it and hang it. Ours are also white -husband used to clean the shower and step out on the colored mat with some of the cleaner still on his feet (who does that?), so we ended up with discolored footprints on the mats). Doesn't happen with white mats.

i love rugs in the bathroom...just redid our master bath and i put decorative wool rugs down...have a 2x3' one in front of shower, but did buy a bathmat to put on top of the rug (tho it would probably be fine without it---i tend to towel dry in the shower, but hub tends to step out dripping wet...)-we just hang it on towel rack when not in use. same thing for the rug in front of tub---tho the tub is so infrequently used that the mat is stored in cabinet.

I def use a clean mat to step onto out of the tub. I don't like to keep it down on the floor because I like having something pristine for that first post-shower step. I don't use a decorative rug for this purpose -- it gets bleached and dried in the dryer. In between "first steps" I hang it like a towel.