The first few palindromic primes in base 10 are 2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727 these are listed in A002385 of Sloane's OEIS.

In binary, all Mersenne primes are also palindromic primes, and the same is true of Fermat primes. Some other binary palindromic primes are 73, 107, 313, 443, 1193, 1453, 1571, 1619, 1787, 1831, 1879.

In factorial base, the most significant digit $d_k$ of a palindromic number has to be 1, thus a prime $p$ must fall in the range $k! < p < 2k!$ or else it is not a palindromic prime in factorial base. The first few factorial base palindromic primes are 3, 7, 11, 41, 127, 139, 173, 179, 191, 751, 811.