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 https://arxiv.org/abs/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 https://arxiv.org/abs/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.
