One of the prohibitions of Shabbat is Horesh, which means plowing or making a furrow. The question arises whether it is permissible for somebody to wheel a baby carriage on soft earth on Shabbat, such as in one's yard after a rainfall, when the earth is very soft. Assuming there is an Eruv, such that the prohibition of carrying in a public domain does not apply, may one wheel a carriage on the soft earth, given that the wheels will likely make furrows in the ground?
The Mishna in Masechet Betza (23) records Rabbi Yehuda's comment (listen to audio for direct citation), "All utensils may not be dragged, with the exception of a carriage, because it presses." This means that when one pulls a carriage, the wheels do not cause earth to be lifted and moved, but rather presses the earth and flattens it into the ground. Horesh, plowing, means lifting earth to create furrows, rather than pressing earth deeper into the ground, and therefore wheeling a carriage on Shabbat does not violate the transgression of Horesh. By the same token, it is permissible for a woman to walk with high heels on soft earth on Shabbat, because this, too, merely presses the earth deeper in the ground, rather than lifting it out of the ground. But dragging other items on soft earth on Shabbat, such as a bed, is forbidden, since the bottom of the bed will lift earth and make furrows in the ground.
Chacham Ovadia Yosef, in Yechaveh Da'at (2:52) extends the Mishna's Halacha even to one who pushes a carriage in a zigzag fashion, which will, in fact, lift earth from the ground. This is nevertheless permissible, Chacham Ovadia writes, because this type of "plowing" deviates from the normal manner of plowing and the individual has no intention to lift the earth as he pushes the stroller.
Summary: One may wheel a stroller on soft earth on Shabbat in a place where there is an Eruv, even in a zigzag fashion. Likewise, it is permissible to walk with high heels on soft earth on Shabbat. One may not, however, drag other items – such as a bed – on soft earth on Shabbat, since doing so makes a furrow in the ground, in violation of the prohibition of Horesh.