Interestingly, there doesn't seem to be a pattern as to the number of
happy numbers in a given range as the base changes:

(My results...might be wrong. :)
How many Happy numbers between 1..10000?
Base 2: 10000 happy numbers.
Base 3: 1988 happy numbers.
Base 4: 10000 happy numbers.
Base 5: 2571 happy numbers.
Base 6: 645 happy numbers.
Base 7: 162 happy numbers.
Base 8: 549 happy numbers.
Base 9: 627 happy numbers.
Base 10: 1442 happy numbers.
Base 11: 196 happy numbers.
Base 12: 24 happy numbers.
Base 13: 582 happy numbers.
Base 14: 93 happy numbers.
Base 15: 164 happy numbers.
Base 16: 2585 happy numbers.
Base 17: 253 happy numbers.
Base 18: 4154 happy numbers.
Base 19: 3647 happy numbers.
Base 20: 1616 happy numbers.
Base 21: 45 happy numbers.
Base 22: 17 happy numbers.
Base 23: 19 happy numbers.
Base 24: 9 happy numbers.
Base 25: 519 happy numbers.
Base 26: 377 happy numbers.
Base 27: 279 happy numbers.
Base 28: 6 happy numbers.
Base 29: 1730 happy numbers.
Base 30: 5266 happy numbers.
Base 31: 11 happy numbers.
Base 32: 11 happy numbers.
Base 33: 84 happy numbers.
Base 34: 192 happy numbers.
Base 35: 77 happy numbers.
Base 36: 50 happy numbers.