Results on the difficulty of specific random 3-SAT problems?
Research has concentrated not on the number of satisfying assignments, but on the clause density $\alpha$. It is (more or less) known that:


Below a certain threshold, the problem is easy. Moreover, the solution set is "continuous" (with high probability), in that all solutions are (more or less) connected.
After a certain threshold, there are no satisfying assignments (with high probability).
Between the two thresholds, the instance is satisfiable (with high probability), but a satisfying assignment is conjectured to be hard to find. Moreover, the solutions belong to exponentially many small clusters that are "far away" from each other: any path connecting them goes through an assignment that satisfies a relatively small fraction of clauses.


These thresholds have been calculated by physicists, but their work is not mathematically rigorous. Rigorously much is known, especially for random $k$-SAT and similar problems for large $k$, but the actual behavior for 3SAT hasn't been rigorously proven.

The literature on the area is enormous, but unfortunately I can't find a recent survey, though these probably do exist. You can start by reading a very short, relatively recent survey by Coja-Oghlan. A recent important rigorous result is Ding, Sly and Sun, who prove rigorously that the unsatisfiability threshold for $k$-SAT exists for large enough $k$.