Answer: The Gemara (Moed Katan 2b) writes that there is a machlokes as to whether one who waters plants on Shabbos transgresses the melacha of choresh, plowing, or zorea, planting. Rambam (Shabbos 8:2) writes that it is considered zorea. Thus, one mustn’t turn on a sprinkler on Shabbos.
The Shulchan Aruch (OC 252:5) writes that one is allowed to open a flow of water before Shabbos that will run onto a garden on Shabbos (See Shabbos 18a). The Rema adds that if the action creates a noticeable sound (avsha milsa), such as a flour mill, then it must not operate on Shabbos.
While the sprinkler may be seen, the Shulchan Aruch Harav (OC 265:8) notes that chazal were not concerned that people will jump to the wrong conclusions and think that such melachos may be operated on Shabbos.
R’ Eliezer Waldenberg (Tzitz Eliezer 4:31; 5:6:3) writes that one may set one’s sprinkler to run on Shabbos, comparing this to switching lights on before Shabbos. While they can clearly be seen, avsha milsa only applies to sound. While one can also switch the taps off, he quotes the Chazon Ish who writes that one must be careful when doing so if there is more than one sprinkler to ensure that they don’t cause the water pressure to increase in the other, thereby causing extra watering. Thus, one must switch it off at the main tap (See Shemiras Shabbos Kehilchasa 8:228).
Nonetheless, R’ Ovadia Yosef (Yalkut Yosef, Shabbos 252:2) and R’ Zvi Pesach Frank (Tzitz Eliezer 5:6:3) write that this isn’t such a concern.
In conclusion, one may place sprinklers on a timer to water one’s lawn on Shabbos, and switch the taps off as necessary.