Drawing graphs with few "sharp" vertices?
It is possible to construct 3-regular planar graphs with $\Theta(n)$ biconnected components (see e.g. fig.16 of this paper), each of which must contain at least one sharp vertex.

On the other hand, if you require higher levels of connectivity, you can avoid having many sharp vertices. In particular, if you have a 3-connected planar graph, it can be drawn (e.g. by using Steinitz' theorem to find a polyhedral representation and then forming a perspective projection) in such a way that all faces are convex, which causes only the outer face to be sharp. But every 3-connected planar graph can be embedded in such a way that the outer face has at most five vertices (the worst case being a dodecahedron) so you can draw every 3-connected planar graph (3-regular or not) with at most five sharp vertices.