If a person eats on Shabbat a salad that contains, for example, lettuce, tomatoes and onions, and he does not like onions, he may not remove the onions from the salad. Halacha forbids removing on Shabbat Pesolet (an undesirable substance) from Ochel (a desirable substance), and this applies even to a "relative Pesolet," a food which is perfectly edible but one does not want to eat. Therefore, the person in this case would have to take the lettuce and tomatoes from the salad, rather than removing the onions.
However, if there is somebody else at the table who enjoys eating onions, one may remove the onions from his salad to give them to that other person. Since he removes the onions not to discard them, but rather for somebody else to eat, this is considered separating Ochel from Ochel, which is entirely permissible on Shabbat. This is the ruling of Hacham Ovadia Yosef, in Hazon Ovadia Shabbat (vol. 4, p. 188, in the annotation).
If some pieces of lettuce in the salad are spoiled, one may not remove them from the salad, as this would constitute separating Pesolet from Ochel. This is the ruling of the Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909), in Parashat Beshalah (2:6; listen to audio recording for precise citation). However, the Ben Ish Hai adds, if one is taking leaves from a head of lettuce, and the outer leaves are spoiled, he may remove them in order to access the fresh leaves underneath them. Just as one may remove a peel of a fruit in order to access the edible part of the fruit, similarly, one may remove the spoiled lettuce in order to access the desirable leaves. Since this is done in order to facilitate eating, it is permissible and does not violate the prohibition of Borer.
Hacham Ovadia Yosef (ibid. p. 191) rules that it is permissible on Shabbat to check leaves of lettuce for bugs, and if one finds a large bug, he may remove it. Since the bug is not mixed with the lettuce, but rather constitutes an independent entity, removing it from the leaf is not considered "separating." Nor is this forbidden on the grounds of Mukseh, because it is permissible to remove an item normally considered Mukseh if this is necessary to access food. Hacham Ovadia notes in this context the Halacha permitting removing a bug from a cup of wine on Shabbat (under certain circumstances).
The Rama (Rabbi Moshe Isserles of Cracow, 1525-1572) writes (302:1) that it is permissible on Shabbat to remove feathers from ones garment, as this does not constitute "separating." Similarly, if pieces of food fall onto ones beard as he eats, he is permitted to remove them from his beard, as this is not regarded as "separating."
Summary: One who is eating a salad may not remove the vegetables that he does not want to eat, unless he does so to give them to somebody else who wants to eat them. One may not remove spoiled pieces of lettuce from a salad on Shabbat, but one may remove spoiled leaves from a head of lettuce if this is necessary to access fresh leaves. It is permissible to check leaves of lettuce for bugs on Shabbat, and to remove bugs that are found.
