As the title says, what are the most popular design solutions when creating a new game or a new game engine?
Patterns were designed to solve a certain problems within a/several domain(s). There are tons of design patterns for games. There is even the possibility that you might come up with your own! ( Well that depends on how much of problem solver you are ).
Allot of articles on creating games implicitly show you certain design patterns.
But ...hey google is your friend. Here
These are pretty much common
But there are so many others that apply as well.
I think the reason you got a downvote is because it's too hard to give you an exact answer. It really depends on what your game specifics are.
Also I agree with lorancou. Don't try to hammer on design patterns. There are other solutions to any problem.