It's probably useful to compare the other questions recently asked here by Sara, which are sometimes stimulating but typically not well formulated or motivated: for example herehere. Like the other questions, this one seems to be asking a very broad question which isn't easily answered for all simple groups. Here the number of Sylow subgroups is measured by the index of the normalizer, which itself requires a lot of case-by-case study. There certainly aren't any convenient tables of results to consult, though the various references given here and in other answers are a good start.
Leaving aside the alternating groups, there are two distinct questions for simple groups of Lie type: When the prime is the defining one, the internal structure of the group given by the BN-pair is a good guide to the normalizer of a Sylow subgroup (characterized in terms of algebraic groups via the unipotent radical of a Borel subgroup). But for other primes there are more subtle questions raised, which many people have studied in the setting of block theory and Deligne-Lusztig character theory. In the literature, it seems most natural to focus on the behavior of each prime according to which factor of the order polynomial for the group it divides. This work aims mostly at comparing the ordinary and the modular representation theory for the given prime, but I'm not sure how explicitly it involves the normalizer order or index wanted here. As Derek Holt points out, there may be extra detail available case-by-case in the case of finite classical groups.
Anyway, it's better to start with a more fully developed question which might get a precise answer.
