can you wash dishes on shabbat?
If the dishes are required for use on Shabbos, it is permitted to wash them.
According to some authorities it is permitted to wash dishes even for the purpose of having the house and kitchen clean and tidy.
Dishes must not be washed with sponges that involve a concern of squeezing out water. Synthetic (non-absorbent) material can be used, or just use your hands. It is permitted to use detergent.
The Shulchan Aruch (OC 323:6) and the Mishnah Berurah (323:26) write that it is prohibited to wash dishes if they are not required for used on Shabbos. However, if the household norm is for the sink to always be clear of dishes, and the sink being full of dishes is seen as clashing with the honor of Shabbos (or Yom Tov), then it is permitted to wash the dishes.
The source for this leniency is Rashi (Shabbos117b), who writes that the prohibition of washing dishes is on account of preparation for weekdays. This implies that if the washing is not for preparation for weekdays, the act would be permitted. [There is certain debate about the position of Rambam in this matter.] The Maharshag, Orach Chaim, Vol. I, no. 61, writes explicitly that this is the case, and permits washing dishes for the sake of hygiene and cleanliness; Tzitz Eliezer, Vol. 14, no. 37 concurs with the ruling.