Moduli of instantons on Calabi-Yau threefolds appear as singlet scalar fields in four-dimensional heterotic GUT and Standard Model theories. Various physical quantities, e.g. Yakawa couplings and mu-terms, depend on them. We explicitely evaluate the number of moduli for a large class of instantons on certain elliptic Calabi-Yau threefolds. In particular. we present an explicit expression for the minimal number of such moduli. We also discuss non-perturbative physics of the instanton moduli as well as generalizations of the presented methods for other Calabi-Yau compactifications and F-theory.