Several different models of random topological spaces have been studied by now, including several models of random simplicial complex.

Andrew Newman and I recently showed that random 2-dimensional hypertrees (Q-acyclic complexes) are aspherical. https://arxiv.org/abs/2004.13572

This is the first case I am aware of where a random space is shown to apsherical with high probability. It is based, in part, on earlier work of Costa and Farber, who showed that for a wide range of parameter, the Linial-Meshulam random 2-complex (with complete 1-skeleton and independent 2-dimensional faces) is *almost* aspherical, in the sense that if you delete one face from every tetrahedron boundary, you get an aspherical complex.