As an honest question (probably with some subjectivity), how many smooth oriented 4-manifolds are actually symplectic? Can I say half? I ask this question because every smooth oriented 4-manifold admits a near-symplectic form, i.e. a closed 2-form which is symplectic away from a finite set of circles.

Some results that might push the percentage one way or the other:

1. Gompf has shown that any finitely presented group can be realized as the fundamental group of a compact symplectic 4-manifold.
2. The Seiberg-Witten invariants are nonzero for symplectic 4-manifolds, and in a sense show that they are the "irreducible" basic forms of smooth 4-manifolds.
3. Every compact symplectic 4-manifold is a branched cover of $\mathbb{C}P^2$.