Random links and $3$-manifolds
There is no universally accepted model of random three-manifolds (or random knots/links) for that matter, however, hyperbolicity is pervasive in all known models. The most popular (but not really satisfying) model of three-manifolds is the Dunfield-Thurston model:
Finite covers of random 3-manifolds.
Nathan M. Dunfield, William P. Thurston.
Invent. Math. 166 (2006) 457-521
and a random manifold in this model was proved hyperbolic by Joseph Maher. Lots of related results (and models) can be found in my preprint
Rivin, Igor.
"Statistics of Random 3-Manifolds occasionally fibering over the circle."
arXiv preprint arXiv:1401.5736 (2014).
There are no hyperbolic split links (they are not irreducible)
It is not clear that a random knot is hyperbolic in all models. In my favorite model of random knots, there is a finite probability that the knot is the trefoil. I am not sure that there is any popular model of links, but it is pretty clear that in any reasonable model a random link will not be split.
The answer to the question "what does a random 3-manifold/knot/link look like?" definitely depends on the model. Here are a few references to complement Igor's answer:
This is the paper where Joseph Maher proves that in the Dunfield-Thurston model (based on Heegaard splittings) a typical 3-manifold is hyperbolic:
Joseph Maher. Random Heegaard Splittings http://front.math.ucdavis.edu/0809.4881
Here is a more recent article with applications to non-random questions:
Alexander Lubotzky, Joseph Maher, Conan Wu. Random methods in 3-manifold theory http://front.math.ucdavis.edu/1405.6410
By contrast, there is a (perhaps more naive) model for generating random knots using the Gaussian random walk in Euclidean 3-space. Surprisingly, such a knot is a satellite knot (hence nonhyperbolic) with positive probability:
D. Jungreis. Gaussian random polygons are globally knotted. J. Knot Theory Ramifications 3 (1994), 455–464.
As others have indicated, there are many different notions of random 3-manifold or random link. Here are two other types of models of random linking:
The Petaluma Model (http://front.math.ucdavis.edu/1411.3308). This model is based on link diagrams with a single multicrossing, with the randomness given by a choice of random permutation, which determines the heights of the arcs relative to one another. They're actually able to compute the distribution of (appropriately scaled) linking numbers on the nose with this model. I don't think that this has been done with any other model. They do some computations of some of the moments of other low order finite type invariants too.
The random projection model (http://front.math.ucdavis.edu/1602.01484). This model starts with a fixed embedding of some circles in some high-dimensional Hilbert space, and randomly projects these onto a 3-dimensional subspace. In principal, the moments of the linking numbers ought to be computable. This is an intriguing model because there are continuously many parameters. It's possible that by varying the initial embeddings, these models can be made to limit to other types of models. Maybe that could explain some of the universality observed experimentally and discussed in the Petaluma paper.
As far as I'm aware, no one has examined hyperbolicity in either of these models.