What cultures have had a lasting effect on Sicilian culture?

Let me do my best to attempt an answer here...

Sicily and its people have always be oddly unique. Italian, yet not Italian. They share much in common with their mainland Italian neighbours (particularly the southerners), in language, culture, history, and ethnicity -- but their particular history stands out more than most.

Widely speaking, the people and culture of Siciliy are something of a great hybrid or "melting pot". Ethnic/genetic studies have been conducted that show a large degree of affinity with southern and even central Italians. (Apologies, I lack the source for this presently.) There are also notable, albeit minor admixtures from principally North African and Near-Eastern peoples. This is undoubtedly thanks to Moorish raids and partial Arab conquests during the early Middle Ages. On average, I think it is fair to say that Sicilians show greater genetic diversity than most Italians -- it is not uncommon for them to have particularly dark complexions or Middle Eastern features (thanks to Arab and Moorish rule); likewise reddish or light hair is not overly rare (inherited from the Norman conquests undoubtedly). Saying that, a plurality if not majority of the population descend from the Roman and pre-Roman inhabitants above all. Undoubtedly there is also a significant proportion of Greek blood in Sicilians, from the time of the Magna Graecia colonies to the Byzantines, but this is somewhat harder to distinguish visually or scientifically thanks to the strong similarities and continual interaction between Greek and southern Italian/Sicilian populations.

Culturally, it's hard to untangle the origins and complex nature of Sicily. At least from my position. There are indeed whole books written on the subject. Of particular interest is the recent history (past two centuries), where Sicily came out of feudal/baronial rule, and eventually joined under Italian Reunification. The process was a tumultuous one, especially considering the poverty of the island that exists even to this day. There were numerous quite apparent causes for the explosion in crime during these 19th century transformations -- the origin of the Mafia/Cosa Nostra in Sicily/Italy and ultimately the USA owes to these upheavals.

Now, while the Spanish (Aragonese and then Hapsburgs) ruled the island for centuries beginning in the late Middle Ages and Early Modern era respectively, the Spanish cultural (or for that matter genetic) influence was probably not very great. It was mainly manifested in terms of aristocratic rule and pan-European wars, I believe. Sicily has an unfortunate (at least in some ways) history of being under foreign domination, both of the involuntary and requested sort (that ultimately leads to loss of too many of their freedoms). The Normans and the Spanish did not secure their rule over the island until they were effectively invited in by certain powerful persons at the time.

One of most intriguing points of Sicilian history is that when the Normans initiated their rule, they did their very best to stamp out all signs of Arab/Muslim culture that had come to dominate the island strongly for a couple of centuries at the end of the 1st millennium. The Muslims and Jews were expelled at various points leading up to the end of the Reconquista. Since the fall of the Roman Empire, Sicily had transitioned between Byzantine and Arab rule for the most part, and the population was fundamentally Orthodox Christian. Only when the Normans came did Roman Catholicism get truly embedded, while Islam was all but eradicated. Norman/Angevin rule did not prove too stable (it was often exploitative or even cruel), and gave way to their Aragonese "liberators", whose rule persisted in some form (through their descendents) until the early 19th century. The Sicilian dialect of the Italian language (some consider it a separate language) and for the most part their culture persisted with little alteration throughout these centuries. Commerce links with mainland Italy were usually present, as indeed Sicily had been a great source of agricultural wealth since pre-Roman times.

So, while this provides a far from conclusive answer, hopefully I've offered a few insights into my limited knowledge of Sicilian history and its influences. Culture is invariably a hard thing to quantify, but perhaps you get the gist anyway. Sorry I can't be any more precise!