To define a random $n$-manifold you typically need to define a complexity on the set $\mathcal M_n$ of all $n$-manifolds you want to consider, which satisfies a finiteness property: for every $k$, there are only a finite number of manifolds having complexity at most $k$. There are various ways to do this in a combinatorial framework.
The minimum number of simplexes $t(M)$ in a triangulation of $M$ is a natural example. I don't think there is much known on random manifolds in this conextcontext. Since $t$ is roughly subadditive on connected sums, a random manifold is might be a connected sum of many manifolds and is would hence be far from being aspherical. In dimension 3 one may restrict to irreducible manifolds. It might seem reasonable to expect that in this conext a random 3-manifold is hyperbolic, but this is still unknown. The first segment $t(M)\leq 11$ shows up a huge number of graph manifolds, see the tables here and here. The hard point here is that it is very difficult to estimate such complexity from below, even for simple manifolds like lens spaces. As an example, the number of hyperbolic manifolds having complexity smaller than $t$ grows more than exponentially with $t$, and the number of lens spaces is conjecturally roughly $2^t$. However, I think we are not yet able to say that the number of lens spaces does not grow more than exponentially. Another fact that shows our ignorance in this conext is the following: we still don't know if the number of triangulation of the three-sphere grows exponentially with the number of tetrahedra, see Gromov's recent questions.
As pointed by Jean-Marc Schlenker, there is another natural complexity in dimension 3 which is easier to treat and gives various nice results: we can consider the smaller set $\mathcal M_3^g$ of all 3-manifolds decomposing into two genus-$g$ handlebodies. A manifold there is determined by an element of the mapping class group MCG of the surface $\Sigma_g$ of genus $g$, and after fixing a set of generators for MCG we can define the complexity of one such element as the minimum lenght of a word which represents it. Many results have been obtained in this context by Nathan Dunfield (with D. and W. Thurston, and H. Wong), Juan Souto and others.
In dimension 4 there are various combinatorial notions of complexity one may use, but they are difficult to treat. One can use Kirby diagrams to represent 4-manifolds and define a complexity by counting the number of crossings in the diagrams as I did here. It is very easy to produce doubles of 2-handlebodies in this context (draw a random diagram and add small 0-framed unknots encircling every component) and to perform blow-ups, so I suppose that in this context most manifolds are of this type: these manifolds are never aspherical and have simplicial norm zero. In this context, Auckly has proved that there is a big discrepancy between homeomorphic and diffeomorphic classes of manifolds: the number of simply connected manifolds of complexity up to $n$ seen up to homeomorphism grows like $n^2$, whereas the number of simply connected manifolds seen up to diffeomorphism grows more than polinomially.