The Mishna in Masechet Shabbat (143) forbids using a sponge on Shabbat, due to the prohibition of "Sehita," squeezing liquid from a material in which it had been absorbed. Thus, ordinary sponges may not be used or handled on Shabbat.
Today, however, special, synthetic "Shabbat sponges" are available, which are made from plastic and do not absorb liquid. One may use such a sponge on Shabbat, on condition that there is some distance between the fibers. If the fibers are tightly-packed, then although the sponge cannot absorb liquid, it nevertheless closely resembles an ordinary sponge, and using this sponge would thus be forbidden Mi'de'rabbanan (by force of Rabbinic enactment). If there is space between the fibers, however, the sponge does not resemble an ordinary sponge at all, and it would therefore be permissible to use it for cleaning utensils on Shabbat.
This Halacha applies to bottlebrushes, as well. If the bristles are densely arranged, without any space between them, this brush gives the appearance of a sponge and may not be used on Shabbat. One may use a bottlebrush if the bristles are separated a bit from one another, such that the brush does not resemble a sponge.