  Following Matveev, a k-normal surface in a triangulated 3-manifold is a
generalization of both normal and (octagonal) almost normal surfaces. Using
spines, complexity, and Turaev-Viro invariants of 3-manifolds, we prove the
following results: 1) a minimal triangulation of a closed irreducible or a
bounded hyperbolic 3-manifold contains no non-trivial k-normal sphere; 2) every
triangulation of a closed manifold with at least 2 tetrahedra contains some
non-trivial normal surface; 3) every manifold with boundary has only finitely
many triangulations without non-trivial normal surfaces. Here, triangulations
of bounded manifolds are actually ideal triangulations. We also calculate the
number of normal surfaces of nonnegative Euler characteristics which are
contained in the conjecturally minimal triangulations of all lens spaces.
