Consider the NP-hard problem of, given a simple graph G, to find a planar subgraph of G with the maximum number of edges. This is called the Maximum Planar Subgraph problem and the best known approximation is (4/9) and is obtained by sophisticated Graphic Matroid Parity algorithms. Here we show that applying a local optimization phase to the output of this known algorithm improves this approximation ratio by a small epsilon > 0. A second local optimization algorithm achieves a tight ratio of (5/12) for Maximum Planar Subgraph without using Matroid Parity. We also show that applying a greedy algorithm before this second optimization algorithm improves its ratio to at least (91/216) < (4/9).