I wondered whether there were an infinite number of palindromic primes written in binary (11, 101, 111, 10001, 11111, 1001001, 1101011, ...) and quickly discovered that it is unknown (OEIS A117697). Indeed, even though almost all palindromes in any base are composite, whether there are an infinite number of palindromic primes in any base is unknown (Wolfram article).
Earlier (in the MO question, "Why are this operator’s primes the Sophie Germain primes?"), I learned that it is unknown if there are an infinite number of Sophie Germain Primes. In addition, is not known if there are an infinite number of Mersenne primes, Fibonacci primes (OEIS A005478), Wilson primes, Cullen primes, not to mention prime twins, quadruplets, sextuplets, and $k$-tuples. No doubt this list of our ignorance could be extended.
It appears to the naïve (me) that there is no nontrivial restriction on the primes for which we know there remain an infinite number in the restricted sequence.
Q1. Is this superficial perception in fact true?
Q2. If so, is there any high-level reason why it is so difficult to prove these statements? Or is each difficult for its own idiosyncratic reason?
I ask this out of curiosity, without expert knowledge of number theory. Thanks for enlightening me!
Questions Answered. Thanks for the wonderfully rich and informative answers! Essentially both questions have been answered: My superficial perception (Q1) is not in fact accurate, as detailed in the examples provided by quid, Anthony Quas, and Joël, augmented by comments by several. A high-level reason (Q2) explaining the difficulty in the examples I listed was nicely encapsulated by Frank Thorne, enriched by appended comments. Thanks!