The City of Houston had an ordinance making it illegal for anyone to "interrupt" a police officer in the execution of his or her official duty. When Raymond Wayne Hill came upon a scene where officers were talking to a pedestrian in the middle of the street, he repeatedly yelled at them, "Why don't you pick on somebody your own size?" He was arrested for violating the ordinance.
After being acquitted at trial, Hill filed a civil suit. When the case reached the U.S. Supreme Court, the ordinance was ruled unconstitutional. The Court said that the guarantee of free speech requires law enforcement officers to tolerate objectionable speech that does not actually obstruct the performance of their duties.
The Court's opinion read, "The First Amendment protects a significant amount of verbal criticism directed at police officers. 'Speech' is often provocative and challenging, but it is nevertheless protected against censorship or challenge, unless shown to be likely to produce a clear and present danger of a serious substantive evil that rises far above public inconvenience, annoyance, or unrest. The Constitution does not allow such speech to be made a crime. The freedom of individuals verbally to oppose or challenge police action without thereby risking arrest is one of the principal characteristics by which we distinguish a free nation from a police state." (Houston v. Hill)
As the dissenting opinion pointed out, under other circumstances, "An individual, by contentious and abusive speech, could interrupt an officer's investigation of criminal conduct," and the First Amendment would not preclude an arrest. For example, if you had just arrived at the scene of a recent robbery and were trying to obtain a description of the perpetrator from the victim to be broadcast to other officers in the vicinity, a bystander's constant interruptions with yelled epithets could delay you in getting timely information that was needed to aid in the perpetrator's apprehension. The bystander's speech would not be constitutionally protected in such circumstances, because the speech—regardless of content—actually delayed and obstructed your performance of official duty.
But the rule of Houston v. Hill means that you must act with considerable discipline and restraint when loudmouths try to demean and upset you with offensive language and gestures. Though restraint is not easy, the cost of giving in to the temptation to show the loudmouth who's the boss can be costly.