The short answer: common for elites for one specific language. The long answer: your question is very broad.

So the first problem in answering the question is that muslims came to be very geographically widespread, and their geographic diversity was well represented in their languages. So when we talk about the languages that muslims knew, we're necessarily talking about very different ethnic groups with different native languages spanning from the Iberian Peninsula to India by the end of the referenced period. It is also worth noting that some of the muslims of the middle ages WERE European and therefore spoke European languages natively.

The second problem is that the "middle ages" is a European concept that doesn't fit well in Islamic history. We use the term "middle ages" because it describes a common set of circumstances and cultural happenings that transpired in Western Europe following the fall of the Roman Empire. Such a cultural and institutional transition did not take place in the Middle East. And so where we might find 300 years of cultural and institutional continuity in Europe, the same period in the Islamic world may cover huge changes that would change the answer to the question depending on the precise period you choose to look at.

The third main issue is that the period typically described as the middle ages (From around 500CE to 1500CE) starts before the rise of Islam. So for the purposes of answering the question, I will obviously start after Muhammad.

The last minor problem with the question is that I'm not sure whether you mean "European languages" in the phylogenetic sense (which would include Persian for example), or whether you mean specifically "languages spoken on the continent of Europe" regardless of whether or not they are related.

Several major linguistic changes occurred after the rise of Islam. For one, Arabic, which was previously limited to the Arabian peninsula begins to spread. This starts very slowly at first because the first Islamic conquerers did not encourage conversion to Islam (which was a major impetus for the spread of the language of the Qur'an). Before the spread of Arabic, Aramaic and Greek were the most widely known languages of the Middle East. Aramaic would begin its long decline (which continues to this day) after the rise of Islam. Egyptian and Berber would also begin their declines into near extinction. I bring this up merely to point out the primacy and importance of the Arabic language. In fact, essentially the only area in the Islamic world to successfully resist this was Persia (and later the Ottomans), which preserves its Indo-European language to this day, but still adopted the Arab writing system. Knowledge of Arabic by Persians and vice versa was relatively common. So I would consider this to be one of the larger examples of muslims learning a European language (depending on your definition as mentioned above).

There were two other European languages that muslim elites studied extensively: Sanskrit (and its daughter languages such as Hindi) but especially Greek.

Particularly during the Abbasid Caliphate, Greek philosophy was extremely important in the Islamic world. Aristotle and other Greek philosophers were studied and discussed extensively. Knowledge of the Greek language was thus relatively common among the educated and merchants who had business with the Byzantines. This was the period of the Faylasufs, who studied "falsafa" (from the Greek Philosophos). This period in Islamic history cements Greek as the primary European language that muslims would know.

This period came to be vital to the European renaissance as muslims preserved much of the knowledge of ancient Greece (and India) for Europeans to later study when the renaissance pushed people to learn Greek in addition to Latin for the first time since the Romans.

There are other more obvious languages muslims would've learned. The muslims of Spain spoke primarily Mozarabic, which was a Romance language.

The muslims of Malta would've been familiar with Sicilian and dialects of Italian, and the semitic language spoken there today (no longer considered merely a dialect of Arabic because of the extensive changes) still shows those traces.

Lingua Franca (the original), also known as Sabir, was also widely used by muslim merchants of the mediterranean, though you could very reasonably argue that this is not a full language, and only partially European.

In summary, Greek was the most studied and respected European language by muslims during the middle ages. It was studied from Iberia to Persia. If we include European languages that were not spoken within Europe, I would add Persian and Sanskrit (and its daughters) as also being extremely important and well-studied.

But there were other European languages that had a more local influence such as in Spain. But perhaps notably, Muslims of the middle ages did not assign much influence to Latin or its daughter languages. And Germanic and Slavic languages are almost entirely undocumented in Arab sources.