The Shulchan Aruch Harav writes (based on a Yerushalmi) that if one who spits into the wind, and the wind spreads it out, he violated the Shabbos prohibition of Zora (winnowing). He adds that it may be only a Rabbinic prohibition, as it is a Melacha Sheina Tzricha Legufa.
However, in Hilchos Pesach he writes that in order for one to violate Zora one must do it in a way of Borer. In other words, it must be similar to classical winnowing where the wind separates the chadd from the wheat.
However, the Ktzos Hashulchan writes that even according to the lenient opinion in the Shulchan Aruch Harav in Hilchos Pesach, there is still a Rabbinic prohibition to spit into the wind (or to destroy Chametz on Pesach by crumbing it and throwing it into the wind) as it appears to winnowing.
However, the Mishna Brura rules leniently for two reasons.
The first is a Tshuva of R' Akiva Eiger where he was asked if one is permited to throw water out of a window where the wind will scatter the droplets. R' Akiva permitted it for two reasons:
The Yerushalmi isn't mentioned in the early poskim and the Halacha is like the Bavli that Zora requires Borer (a separation of good from bad) and
Even if there is a concept of Zora where there is no Borer, it's possible that it applies only to things that grow from the ground (as winnowing is similar to threshing and gathering which are only prohibited to be done to items which grow from the ground). As R' Akiva Eiger says that spit is considered something that grows from the ground, the Yerushalmi prohibits spitting into the wind. However, water (which) isn't something which grows from the ground and would be permitted to scatter.
One doesn't care if the water is scattered, and according to some opinions a Psik Reisha whose result isn't needed is allowed.
The second reason is the Alfei Menashe which says that the Yerushalmi isn't prohibiting spitting into the wind because of Zora but because of carrying in a public domain. Therefore, everyone (including the Yerushalmi) would agree that Zora requires Borer.
So it appears from the Shulchan Aruch Harav that there is (at least) a Rabbinical decree against scattering objects (the Shulchan Aruch Harav includes bread in the prohibition, and the Ktzos Hashulchan includes paper).
According to the Mishna Brura, it would appear to be permitted.
However, according to Rabbi Yosef Yitzchak Zalmanov, one would be permitted to apply spray deodorant on Shabbos. Presumably, it's because there's no wind involved in spraying deodorant.