There are three basic guidelines to avoid the prohibition of Borer and select from a mixture on Shabbat: The Ochel (good) must be selected from the P’solet (waste; It must be selected by hand; the selection must be for immediate use.
The Poskim discuss whether it is permitted to bypass the problem of Borer by scattering the mixture to the degree that it is no longer considered a mixture. Would that allow a person to select even the P’solet from the Ochel? For example, if a person has a cup of mixed nuts, also containing a type of nut he doesn’t care for (which would be the “relative P’solet”), may he scatter the nuts on the table to enable him to pick out the undesired nuts? The Poskim are lenient, based on a Rashi in Masechet Shabbat (p. 74a) who explains that a sage scattered a basket of fruit in front of the people in a manner that there would not be a problem of Borer, as cited by the Sefer Shabbat Shlomo (Rabbi Shlomo Miller, p. 179).
Rabbi Miller also applies this principle to a case where a person has a bowl of Hamin containing meat, potatoes etc. If he does not want the potato, just the meat, he may push it to the side of the bowl to gain access to the desired meat. Since he did not remove it from the bowl, he is merely spreading out the mixture, which is permitted.
Similarly, Rabbi Miller permits a person immersing in a Mikveh on Shabbat to push away an area of water which appears to be dirty. Since he is not removing it from the Mikveh, it is analogous to spreading out the mixture.
It is permitted to break up a mixture by scattering its contents, and similarly one may spread out the contents of the mixture within its container.