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There is a potential for mess of course. Inserting it is no more messy than non-application tampons. Removing them is of course there concern for spillage but just be careful, you'll get use to it really quick. It is really easy to clean. My flow actually seems cleaner and less old as it not drying up in the tampon or pads. Less odour too. It's my first time using but I see the benefits already and I'm prepared for the learning curve, just like the first timing using tampons.
There is less waste for bins so a lot less messier from that side. Actually reminds me of having to change tampons in public toilets and trying to get it into to bin without touching it or trying not to see the contents of it. I think a cup will make it lot easier and reduce the need to use public toilets as you can go longer before emptying and nothing to throw into bins.
I find I can't use tampons any more as my flow has gotten lighter and tampons just dry you out and can be sore if left in too long. So at night the cup is much more comfortable than big pads.
Tampons you should change every regularly 4 to 6 hours. With cups, you only have to change every 8 to 12 hours and you don't have to carry around supplies.
Also with cups you safely insert it when you expect AF unlike tampons.
Tampons leave bits of fibres behind, and there is a TSS risk as well.
And that's just a few of the benefits so far.