Let Q−(3, q) be an elliptic quadric and Q+(3, q) a hyperbolic quadric in PG(3, q). For ɛ ∈ {−, +}, let Tɛ denote the set of all tangent lines of PG(3, q) with respect to Qɛ(3, q). If k is the minimum size of a Tɛ-blocking set in PG(3, q), then it is known that q2 +1 ≤ k ≤ q2 +q. For an odd prime q, we prove that there are no T+-blocking sets of size q2 + 1 and that the quadric Q−(3, q) is the only T−-blocking set of size q2 + 1 in PG(3, q). When q = 3, we show with the aid of a computer that there are no minimal T−-blocking sets of size 11 and that, up to isomorphism, there are eight minimal T−-blocking sets of size 12 in PG(3, 3). We also provide geometrical constructions for these eight mutually nonisomorphic minimal T−-blocking sets of size 12.