Hey! This is an interesting question but one that I want to slightly change the wording to. We cannot speak of an Islamic "clergy" in the same way we may speak of the Catholic clergy. There was no set hierarchy within the Islamic faith; no pope appointing cardinals appointing parish priests. Nor was the Caliph to be considered like the Pope. He was a political ruler who, although at times may have dabbled in religious matters (or seriously attempted to influence them), had no inherent ability to set religious doctrine.

Rather, influence depended on what Wael Hallaq has termed the epistemic-moral authority of the various religious figures in society. Each jurist or other religious figure would exist in a relatively even playing field in which their importance came from the number of people they could convince to respect their religious credentials, not the ranking of the figure who appointed them. Hallaq sums up this attitude in the following quote:

If a Caliph actively participated in legal life, it was by virtue of his recognized personal knowledge of the law, not so much by virtue of his political office of military power. Thus, legal authority in Islam was personal and private; it was in the persons of the individual jurists (be they laymen or, on occasion, caliphs) that authority resided and it was this competencein religious legal knowledge that was later known as ijtihad - a cornerstone of Islamic law.

This lack of formal hierarchy is part of the reason for the existence of several different schools, or madhhabs, of Islamic law. Within Sunni Islam, there are traditionally four main, accepted schools - the Maliki, the Hanafi, the Hanbali, and the Shafi'i. Each of these schools was founded by their namesake who was viewed by his followers as having an expert view of the law and being worthy of recognition and able to set precedent. This is what Hallaq referred to by ijtihad. Ijtihad, in a nutshell, is the ability of a qualified jurist to employ his mental faculties, reasoning, and logical comparison (through a process called qiyas) to reach the conclusion to a question whose answer is not explicitly stated in the Qur'an or Hadith and to set precedence for future cases to be decided upon. Notably, many argue that the "Gate of Ijtihad" was closed between the 9th to 11th century CE, with the Muslim 'Ulema deciding that all issues worth deciding have been decided and that ijtihad was no longer permitted. This has been challenged in recent years but the debate continues and there has been no consensus ^(haha) reached.

Going back to the madhhabs, each of the schools would be dominant in a particular area of the Islamic world and would vary on any number of issues, from marriage to business contracts to punishments. For example, the Ottoman Empire largely followed the Hanafi school of thought, which was more liberal on certain issues such as divorce and contracts. Conversely, much of the Arabian peninsula followed the Hanbali school, known for its strict interpretation of the law.

Merely living in an area where one school was dominant didn't mean you were bound entirely to their interpretation. One could, for lack of a better example, "shop around" for opinions that may benefit your particular case. You could do this both within one school of thought, finding a religious scholar who would issue a fatwa, or religious ruling, sympathetic to your cause. Or, you could reach outside of your school of thought. Women seeking divorces would occasionally settle the matter before the Hanafi school, given their penchant to grant divorces more easily and more commonly.

While this difference of opinions is applicable when talking about certain crimes and court cases, it is important to note that for a small number of crimes in particular, the hudūd crimes (meaning "set" or "fixed"), the punishments were outlined within the Qur'an and variation was much less common. These include crimes such as highway robbery, murder, drinking alcohol, adultery, and unfounded accusations of adultery. Although these crimes had higher standards of proof (nearly levels of absurdity sometimes), their punishment, if guilt was found, would have been more evenly applied across Islamic lands. Thus, one could "shop around" less with these issues than with others. These crimes are also fundamentally based on the Islamic faith, so these represent some of the more "institutionalized" impacts of the Muslim religious authorities upon normal peasants, more akin to the codified Catholic church law.

In sum, it is difficult to compare the apparatus of the Catholic church to the non-Western, non-Christian tradition of Islam. Authority was more fluid, based upon your personal ability to be recognized as a proficient scholar instead of upon the institution that appointed you. A plethora of doctrines and opinions existed simultaneously and few people were bound to follow a single school of thought. In some ways, this allowed for a less-controlling environment than the rigidity of official church doctrine in medieval Christendom. However, certain punishments would be applied relatively more evenly across the various schools of thought and show that, despite the differences in opinion on many subjects, Muslim society was very much based in a certain religious thought that permeated down to the lives of the average man.

I hope that I have given at least the start of an answer! If you have any questions, please ask and I'll get back as soon as I can.