Those type of sponges. Welcome to Daily Halacha. Can one do dishes on Shabbat? The answer is that if you need the dishes for Shabbat—for example, after the Friday night meal or for the Shabbat day meal—and you need the plates, you need the forks, then you're allowed to wash dishes on Shabbat for Shabbat day from Friday night. Or let's say you've finished the Shabbat day meal, but you know you're having guests and need the plates again. Then you're allowed to wash them. For Seudah Shelishit (third meal), if you need the plates or forks, then you're allowed to wash them.
Otherwise, you're not allowed to wash dishes on Shabbat, because if you wash them on Shabbat but aren't going to use them on Shabbat, then we consider that mechin miShabbat lechol. We consider that you're preparing from Shabbat to a weekday. A person is not allowed to halachically prepare anything on Shabbat for a weekday. So, are you allowed to do dishes on Shabbat? Yes, but only if you're going to use those dishes on Shabbat.
Cups, however—glass cups or whatever cups you use—the Gemara says you're allowed to wash them throughout Shabbat. Why? Even though you're not specifically planning to use them for one meal or the other, you're always allowed to wash cups. Why? Because the Gemara says people constantly drink. Since people constantly drink throughout Shabbat, there's a very good reason to assume you're going to need to reuse the cups. So when you wash them on Shabbat, even though you haven't calculated when you're going to use them next, because people drink throughout Shabbat, you're always allowed to wash cups on Shabbat. Anything that's dirty inside, inside the sink. But for all the other utensils, a person has to make sure to wash them specifically for use on Shabbat.
Hacham Ovadia Yosef says, let's say you have a sink piled all the way to the top with dishes from the Friday night meal and the Shabbat day meal, and it might start to smell. The wife—the woman—gets very, very uncomfortable from it, to the point that it takes away her Oneg Shabbat. It takes away her pleasure of Shabbat because the smell of the sink bothers her. Or, even if it doesn't smell, it looks very messy and dirty. Hacham Ovadia Yosef says you're allowed to do the dishes to clean the sink in order to calm the woman or the wife, because otherwise it might take away her Oneg Shabbat, the pleasure of Shabbat, the enjoyment of Shabbat.
See, even though they're not going to reuse the utensils they're cleaning, because they're doing it to avoid tza'ar, to avoid pain on Shabbat from seeing such a mess in the sink, there is a heter (leeway) to clean those dishes. But even when Hacham Ovadia Yosef says you're allowed to clean those dishes, it means you should clean them just enough, very quickly, to take away the feeling of messiness and the feeling of a messy or piled-up sink. But actually sitting and starting to scrub every plate should be avoided on Shabbat unless you're going to reuse those plates.
Now, when it comes to cleaning dishes, you're not allowed to use a sponge because a sponge on Shabbat is sochet. Squeezing on Shabbat is not allowed. The rule for which sponge you're allowed to use is very simple. There are so many different companies, so the rule is very simple: Any sponge that actually takes in liquid, sucks it in or takes the liquid inside the sponge, that will be...
But all the other types of sponges, where the material is too thick, made out of metal or thick plastic, and the pieces are too far away from each other, don't really soak up the water or keep it in the sponge. The water just goes all the way through. Those types of sponges, because the water doesn't get dissolved inside the sponge, aren't really sponges. So those are allowed to be used on Shabbat.
The rule is very simple: A sponge that actually works, where you put water in it and the water stays inside, and you need to squeeze the sponge to get the water out, you're not allowed to use. That type of sponge you're not allowed to use on Shabbat. But a sponge that doesn't really work—which is what people use on Shabbat, basically called a Shabbat sponge—doesn't keep the water inside. The water goes out the other side of the sponge, and you don't need to squeeze it for the water to get out, because the water is already getting out. That type of sponge you're allowed to use on Shabbat to do dishes.