How does the Bill of Rights and the US Supreme Court regulate the police?

Neither Bill of Rights or the US supreme Court directly
regulate the activities of police. The actual regulation of police, or for that matter,
of any government institution or process are regulated by separate laws, rules and
regulations, enacted separately for different fields.

The
main purpose of the bill of rights is to ensure that such laws, rules and regulations do
not violate the bill of rights. For example consider the fist clause of Bill of Rights
forming the first amendment to the constitution. It
says:

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Congress shall make no law respecting an
establishment of religion, or prohibiting the free exercise thereof; or abridging the
freedom of speech, or of the press; or the right of the people peaceably to assemble,
and to petition the Government for a redress of
grievances.

Also the function
of supreme court, in relation to bill of rights is to decide if any action by government
or any law passed by the it violates the bill of rights. This is also done by supreme
court only when referred it following the laid down procedures. In deciding on such
matters, the supreme court also interprets the provisions of law that are open to
ambiguity. For example, the fifth amendment uses the term "due process of law", which
can be interpreted to mean different things by different people. Supreme court
interprets and clarifies such provisions of bills of rights as and when
required.