Could the police bring a dog to school to look for contraband in students' belongings?

Yes. In a 1985 case called New Jersey v. T.L.O., the Supreme Court ruled that where there are reasonable grounds for suspecting that a search will turn up evidence of students' violating the law, searches by school officials are permitted.

Another case held that the use of dogs in searches is not overly intrusive on a person's privacy and is therefore permitted. Putting the two cases together and considering the need for protection of the students who are not involved in criminal activity, the use of search dogs is permissible and is not in violation of the students' constitutional right to privacy.