One of the unique aspects of the Catholic world was the parallel systems of secular and ecclesiastical power. Church authorities, responsible to the Catholic hierarchy going up to the pope, operated independently of the secular authorities answerable to Christendom's kings and princes. The relationship had some aspects that were cooperative, and some that were adversarial.

The Inquisition was a product if this bifurcation in authority in the Catholic world. Although Christendom's secular princes generally outlawed heresy, the inquisition (at least originally) provided an additional system of investigating theological crimes, to some degree independent of the state.

In the Islamic world, while imams certainly did sometimes disagree with and operate independently of the local sultan, there was no large scale, independent ecclesiastical structure like that of the Catholic world. As such, the suppression of heresy fell solely to the "secular" powers.

To what extent these powers actually persecuted heretics such as Shi'ites (in Sunni realms) or Druze varied. The Mamluks in the mid-14th century, for instance, made a concerted effort to outlaw Shi'a Islam in Syria. In the case of the Ottoman persecution of the Druze, on the other hand, it seems as though the Ottomans were more concerned that the Druze wanted some degree of political independence than the mere fact that their religious beliefs were heterodox. The Druze and Ottomans thus found themselves in a cycle of rebellion, punitive countermeasures, and mutual hatred that requires no understanding of Druze theology to understand.

Yes there has, but it is not quite the same as the Catholic inquisition that targeted other religious groups... I would classify the "Mihna" as an Islamic Inquisition, although it was perpetrated by Muslims against other Muslims. Specifically, it had to do with whether the Qur'an is a holy text that was created by the divine in the world and in time, or whether the Quran is an eternal text - the inquisitors, with the authority of the Caliphs Ma'mun and Mu'tassim, forced people to proclaim that the Quran was created. It involved literally putting people to the question: "Do you believe the Quran is created or eternal?" so I think it qualifies as an inquisition - and there are pretty brutal reports of what the enforcers did. The most famous victim of the Mihna was Ahmad Ibn Hanbal, the founder of a still-highly-revelant and dominant school of Islamic law--he became something of a martyr and is arguable responsible for the fact that the regime was forced to back down from 'the question'...the tragedy of the Mihna is also (and this is somewhat IMO, though you can find it in the sources below as well) responsible for the disrepute that 'Islamic rationalism' fell into for centuries (a disrepute which lasts today).

Some references:

Oxford Islamic Studies Online: http://www.oxfordislamicstudies.com/article/opr/t125/e1510?_hi=0&_pos=4388

Old but interesting classic on the topic: Ahmed Ibn Hanbal and the Mihna: A Biography of the Imam Including an Account of the Mohammedan Inquisition Called the Mihna, 218-234 A.H, by William Patton

Decent section in Defenders of Reason in Islam: Mu'tazililism from Medieval School to Modern Symbol, R. Martin and M. Woodward

Also The Formation of Hanbalism: Piety into Power by Nimrod Hurvitz (Hanbal was the most famous victim of the Mihna and his eventual triumph of sorts says a lot about the theological and political divisions exposed by the event, as well as the VERY important lasting effect of this event on the development of Islamic theology and thought.)

In addition has there ever been a Jewish Inquisition

to really have an inquisition in the Spanish sense, you really need to have government and religion mixing, and historically Judaism hasn't often been a governmental religion. Additionally, Judaism never had the same motivation for the inquisition, a large group of suspect believers, never really existed the same way.

However, there are a couple possible parallels. In the Middle East around the 10th century, a group called the karaites became prominent (though their ultimate origin is unclear). They essentially rejected tradition as a source of revelation, thereby challenging most of jewish practice and establishment. But without a government, no measures so draconian could be undertaken.

In early modern Eastern Europe, a religious dispute arose between chassidim and misnogedim. Without getting into the nitty-gritty, these groups had a big beef (pun intended--one of their disagreements involved kosher beef). But, without a government, they resorted to other methods, such as polemics and excommunication.

In ancient Judea the two largest jewish factions were the Sadducees and Pharisees. They fought for the support of the government, and for control of the temple. But, this is more religious war than heresy-fighting.

in none of these is there the pretext of rooting out secret unbelievers. The first two were to get people to renounce a heretical view, and in the last the goal was for the right faction to control the establishment. They're the closest things, really, but not inquisitions.

Your question called into mind something I'd read recently, so I went to look it up. It's a tidbit concerning Muhammad Baqir Bihbahani, aka Vahid Behbahani. Re-reading it, I would hesitate to call this a true Inquisition, but just in case it may prove of interest to you, here is the passage I had in mind:

The second major change of the period was the victory of the Usulis, achieved by the somewhat violent methods of the eminent scholar Vahid Bihbehani (1705–92), who defined the role of ijtihad with great clarity, and made its use obligatory for jurists. Any Shiis who refused to accept the Usuli position were outlawed as infidels, and opposition was ruthlessly suppressed. There was fighting in Kerbala and Najaf, and some Akhbaris died in the struggle. The mystical philosophy of Isfahan was also banned, and Sufism was suppressed so savagely that Bihbehani’s son, Ali, was known as the Sufi-slayer.

Source: Armstrong, Karen. The Battle for God: Fundamentalism in Judaism, Christianity and Islam (Random House, 2001). Google Books link