Not including M51, here's a histogram of the final hexadecimal digit of the exponents of Mersenne primes:

Code:

#
# #
# #
# #
# # # # #
# # # # # #
# # # # # # #
# # # # # # # #
# # # # # # # #
# # # # # # # # #
2 1 3 5 7 9 B D F

We can see that the prevalence of 1 (mod 4) vs. 3 (mod 4) is really a two-to-one prevalence of 1 (mod 8) vs. the other three:

(We include M51 here since its exponent has already been revealed to be 5 (mod 8) )

Code:

1 19
3 9
5 ~~11~~ 12
7 10
2 1

Similarly, a histogram of the final hexadecimal digit of the exponents of Wagstaff primes:

Code:

# #
# #
# #
# # #
# # # #
# # # # # #
# # # # # # # #
# # # # # # # #
# # # # # # # #
1 3 5 7 9 B D F

We can see that the prevalence of 3 (mod 4) vs. 1 (mod 4) is really a two-to-one prevalence of 7 (mod 8) vs. the other three: