In the sense of Baire categories, we prove that the set of pairs of bivariate copulas that are comparable -- in either direction -- under the increasing convex order is nowhere dense in the space of all pairs of bivariate copulas equipped with the uniform metric. As a consequence, a topologically generic pair of bivariate copulas is not comparable under this order. We further extend the Baire-category programme to two additional stochastic orders on the space of bivariate copulas: the bivariate convex order and the stop-loss order on the sum of the components. For each of these orders, we establish that the set of comparable pairs is closed and nowhere dense, and we show that a topologically generic pair of bivariate copulas is simultaneously incomparable in all three orders. These results complement those obtained in [F. Durante, J. Fernández-Sánchez, C. Ignazzi (2022). Baire category results for stochastic orders. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 116, article 188] for the lower orthant order on copulas.