It is my opinion that the following are frivolous and should be immediately discarded:
1. Retroactively protesting that your team was read the wrong bonus, in formats where bonuses are not directly related to tossups
2. Protesting that your own team should not receive points for a correct answer
3. Protesting that an answer that is widely different from the answer on paper is correct, unless the answer on paper is clearly incorrect
4. Counter-protesting the resolution of a protest against your team, or refusing to accept the other team's protest if you know they're right
5. Protesting that the questions are too long, too hard, contain too much academic material, contain not enough math calculation/trash/general knowledge, or contain not enough of an academic subject your team is really good at; that the tournament director and staff is clearly biased against your team (unless this is

or CBI, in which case you're entirely justified in making the protest, but the staff is going to discard your protest as frivolous anyways); or that (an)other team(s)'s player(s) should be disqualified for being "too good" while alleging that they have heard the questions before.
The following are borderline; they should probably be resolved, though the coach/player may look like an idiot. In order of "most likely to get angry stares from competent people" to "least likely to get angry stares from competent people":
1. Protesting that a team's (incorrect) answer should not have been accepted, then bragging about "screwing a team out of a national championship" on an online message board.
2. Protesting that a team is in violation of a minor, stupid rule
3. Insisting that a protest be ruled upon in a blowout game in a format where paper tiebreakers do not exist (or are noted not-predictive tiebreaker head-to-head)
4. Protesting any aspect of moderator discretion (the sole exception to this is if the moderator is acting like an idiot by either wasting time in a timed round or refusing to accept answers that have demonstratively been proven correct)
The following should always be ruled upon, and are incumbent upon a coach to protest:
1. Protesting that a reasonably close answer should be acceptable for the answer on paper
2. Protesting that an incorrect clue caused a player to give an incorrect answer, or that the answer on paper is wrong
Generally, I think that the problem with borderline situation #2 is not with the coach. There are several coaches who insist on a strict reading of the rules, and if they can point to me where a clear violation of the rules occurs, then I have to side with them; I may view them as pains in the neck, but not as "unprofessional." The problem is that the rule is stupid and should be changed. The reason that these kinds of rules reflect badly on the coach is that the same coaches that are tirelessly enforcing these rules are the same coaches the refuse to change the rules. Therefore even though the coaches are right in enforcing the rule, they look unprofessional for not being willing to change stupid rules (because said stupid rules benefit their team, because they are the only coaches calling teams out on violating them).