There are two "wildcard solutions" that each can be used to solve nearly all of the exercises. Due to the no-reuse rule you can use each of them only once, and they're officially used for exercises 3 and 13.
Just explore all kind of scalar references. If you're really desperate you can plow through Perl_sv_2pv_flags() in sv.c in the perl source code to see exactly how references get stringified.
Good candidate for wildcard one: Find a way to override the meaning of the regex.
The pattern of exercise 4 is misleading, but the search down this incorrect path may reveal the solution to exercise 8. Exercise 4 is actually simpler than you'd think.
A fairly recent and unknown class of built-in objects.
Alias. (Last year there was a similar one which could be solved by manipulating $[. I tried to prevent that this year, though initially without success)
Make modification fail.
Find something that fails to get assigned anywhere. See also hint about 4 and 8 above. The ability to acquire this thing at all is most likely a bug.
Subtly perturb the meaning of the regex.
Have $foo change as a result of the first part of the predicate test.
Find a way to scribble onto $| so rudely it never gets properly turned into 0 or 1. This mechanism is most likely a bug.
Find a variable that ignores being assigned to. This behaviour is most likely a bug.
The best spot for wildcard two: A feature was added in 5.8 with which you can modify constants.