At the public u. my son attended, the business school was very competitive for admission (starting junior year) and a couple of the specific high demand engineering majors were very competitive. Also, the undergrad public policy program had become very competitive. However, most majors only required that you completed a few courses in their department with at least a B average in those courses in order to make that department your major.
Nursing is a different story. It is almost always more difficult for admission than for the general student population. For some universities that accept nursing majors for the junior year, admissions can be highly competitive.