Betti number
These surfaces are never algebraic or Kähler. The minimal ones with "b"2 = 0 have been classified by Bogomolov, and are either Hopf surfaces or Inoue surfaces. Examples with positive second Betti number include Inoue-Hirzebruch surfaces, Enoki surfaces, and more generally Kato surfaces. The global spherical shell conjecture implies that all minimal class VII surfaces with positive second Betti number are Kato surfaces, which would more or less complete the classification of the type VII surfaces.