Consider a quadratic cone K in the 3-dimensional projective space PG(3,q) over a finite field of order q, where q is a prime power. Let E (respectively, T, S) denote the set of all lines of PG(3,q) that are external (respectively, tangent, secant) with respect to K. We characterize the minimum size blocking sets in PG(3,q) with respect to the line set A, where A is one of E, T, S, E∪T, E∪S and T∪S.