The relatively-big, relatively-new thing at the moment is the application of SAT solvers to CGoL problems. Donald Knuth got the ball rolling on this, but we're still in the very early days of seeing what is possible with SAT solvers.
Every now and then a lucky or inspired SAT solver problem setup will throw out an answer to a really difficult-looking problem, with no apparent effort. But then that tends to tempt people into setting up more difficult problems to solve... and of course it's still very easy to set up problems that cover such a large search space that the search would take billions of years to complete on a planet-sized supercomputer.
So it's still very much an art form, rather than an exact science, to figure out what searches to try next.
Every now and then a lucky or inspired SAT solver problem setup will throw out an answer to a really difficult-looking problem, with no apparent effort. But then that tends to tempt people into setting up more difficult problems to solve... and of course it's still very easy to set up problems that cover such a large search space that the search would take billions of years to complete on a planet-sized supercomputer.
So it's still very much an art form, rather than an exact science, to figure out what searches to try next.