Students' works for academic classes are protected by copyright—they are 'original works of authorship' which have been fixed in a 'tangible medium of expression'—and must be treated with appropriate respect. While it is likely a fair use for an instructor to retain copies of student work for academic or pedagogical purposes, publishing or distributing copies of those works would likely infringe the students' copyrights. (Note also that disseminating students' works may also violate FERPA protections.)
As instructors, if you wish to publish or otherwise disseminate a student's work, you should obtain the student’s permission in writing to do so, and be specific about the type of publication being made (e.g. display on a website, submission to a journal, entry into student contests), the time frame of the publication, and the purpose of the publication. Where the student will be personally identified with the work, also be sure to obtain the required permissions under FERPA.
One Court of Appeals has ruled that although students possess copyrights in their schoolwork, the use of online plagiarism detectors is a 'fair use' and that no copyright infringement occurs through their use. A.V. v. Iparadigms, LLC, 562 F.3d 630 (4th Cir. 2009). Despite this ruling, any professors wishing to use plagiarism detectors should notify their students of this in the syllabus, with a provision that the submission of any assignments for the course authorizes the professor to use plagiarism detection software.