Many kitchen sinks have a sprayer next to the regular faucet. Someone has told me that she grew up with the knowledge that one may not use such a sprayer on Shabas. I have never otherwise heard of such a prohibition; nor can I think of any reason it should exist. Does anyone know of such a reason, or of an authority who has said not to use (or who has explicitly said one may use) such a sprayer?
From Rabbi Dovid Ribiat's The 39 Melochos (p. 378):
a) Aerosol spray cans
The Igros Moshe permits the use of aerosol cans on Shabbos because the scattering is not caused directly by blowing or wind-force, but rather by the pressure of the liquid as it is forced through the microconduits in the nozzle head. Since the scattering is not directly caused by the wind, it is not similar to Zoreh and is permissible.23
a-1) Faucet sprayers
Based upon the reasoning permitting the use of aerosol cans, it follows that one may use the "spray" attachment of the kitchen sink for washing dishes on Shabbos. Here too, the spray is not created by wind-force, but rather by water pressure forcing water through the small holes in the faucet or head of the spray attachment. (However, it is of course forbidden to use the hot water on Shabbos; only cold water may be used - see Bishul, Chapter V;1).
23 [Translation by Fred:] In the sefer Hilchos Shabbos by Moreinu HaRav Shimon Eider shlit"a, it is written that he heard directly from R' Moshe Feinstein ztvk"l to permit since "the prohibition of Zoreh applies when the wind scatters it, but here, the air contained inside the vessel propels it outside, and the scattering primarily occurs since it exits through a small aperture." I have also seen brought from the responsa Minchas Yitzchak (VI, §26) that everyone agrees that there is not even a rabbinic prohibition of Zoreh here.
In the footnotes in the back, Rabbi Ribiat additionally notes the general leniency mentioned by Rabbi Akiva Eiger (I, §20, cited in the Bei'ur Halacha mentioned in the comments on this question) that scattering water in the wind is probably permissible even according to the stringency of the Rama (OC 319:17, based on the Y'rushalmi Shabbos 49a), who prohibits scattering spit into the wind due to the m'lacha of zoreh (winnowing).