What language(s) was Christ likely to have spoken during his mortal ministry?

The answers to this question will vary according to the various perspectives.

Nevertheless, it is thought by some that Jesus spoke only Aramaic.

St. Thomas Aquinas seems to be of this opinion also:

Christ in His own person purposed preaching to only one nation, namely the Jews. Consequently, although without any doubt He possessed most perfectly the knowledge of all languages, there was no need for Him to speak in every tongue. And therefore, as Augustine says (Tract. xxxii in Joan.), "whereas even now the Holy Ghost is received, yet no one speaks in the tongues of all nations, because the Church herself already speaks the languages of all nations: since whoever is not in the Church, receives not the Holy Ghost." - Question 176. The grace of tongues

Even Pope Francis has spoken on this:

Language is communication. Without language, whether written or spoken or in whatever medium it may be, one is unable to communicate with others in a meaningful, relational way. When we ask how God communicates with humanity, language necessarily plays a role in this inquiry. This is perhaps why there may be such high stakes surrounding the answer to the question What language did Jesus speak?

In the Summer of 2014, during his visit to Israel, Pope Francis found himself in a minor scholastic repartee with Prime Minister Benjamin Netanyahu. The exchange was short, but not without implication: “Jesus was here, in this land. He spoke Hebrew,” Netanyahu said, in connection to the strong relationship between Judaism and Christianity. “Aramaic,” the pope corrected. “He spoke Aramaic, but he knew Hebrew,” said the prime minister, regaining his ground.

First the historical matter. Did Jesus speak Aramaic, Hebrew, or Greek?

It is likely that Jesus’ primary language was in fact Aramaic, as the pope suggested, and as Netanyahu admitted. Aramaic, a Northwest Semitic language, was popularized as the lingua franca of the ancient Near East in the Neo-Assyrian period (745-609 BCE) and was used widely as the language of diplomacy during the Persian period (539-332 BCE). The Hellenistic, Hasmonean, and Roman periods in Palestine all display a use of Greek and Hebrew as well. Hasmonean coinage, for instance, bears examples of each of these languages being used in official minting. Thus when we arrive at the figure of Jesus, a Roman period Palestinian Jew from the Galilee region of the province of Judea, and as we move into the works of the New Testament authors, it is not surprising to find that all three of these languages cooperate together.

Documentary and textual evidence seems to point to the use of Aramaic as a “default” language for communication in Jewish Palestine at this time. While Hebrew may have been favored for biblical, poetical, or liturgical texts (and may have even been used colloquially especially in Jerusalem), Aramaic appears to have been the “low” language, to frame things in sociolinguistic terms. These two registers of language existed simultaneously, in a condition known as diglossia, and Aramaic seems to have been the vernacular. Thus when Pope Francis corrected Prime Minister Netanyahu, he was, in fact, correct. Jesus likely spoke Aramaic, but the prime minister’s caveat also holds water—Jesus knew, and maybe at times even spoke, Hebrew. I would not doubt if Greek were in his repertoire as well, but it is hard to say how much language knowledge a first-century Jewish Galilean mason possessed beyond the two already determined. - Speaking the Same Language

Outside Jesus' mother tongue, he may hay employed other languages, but the Sacred Scriptures are silent on this issue for the most part.

Aramaic

Christ taught and spoke mainly in Aramaic. The Gospels record some of Christ's words in the original Aramaic. When He healed a little girl, He said in Aramaic, “Talitha cumi,” (Mk 5:41) which means, 'Little girl, get up.' When He healed a deaf man, who spoke poorly because he was deaf, He said, “Ephphatha,” which is Aramaic for, “Be opened.” (Mk 7:34). On the Cross, when Christ cried out to His Father in Heaven, He spoke in the language of His daily life on earth, “Eloi, Eloi, lama sabachthani?” which means, “My God, my God, why have you forsaken me?” (Mk 15:34). Christ mainly spoke Aramaic, as was the custom for persons living in Israel and other areas around the Mediterranean during that time period.

Hebrew

Christ also spoke and taught, occasionally, in Hebrew. He often read the Scriptures and taught in the synagogues on the Sabbath. “And he came to Nazareth, where he had been brought up; and he went to the synagogue, as his custom was, on the sabbath day. And he stood up to read; and there was given to him the book of the prophet Isaiah. He opened the book and found the place where it was written…” (Lk 4:16-17). When Jesus read from the Scriptures, He read in Hebrew. The Hebraic Jews of Israel generally knew both Aramaic (their daily language) and Hebrew (the language of their faith). Hebrew was their preferred written language since it was the language of their Scriptures. After reading the Scriptures, Jesus could have taught in either Aramaic or Hebrew and been understood. However, He probably taught in the synagogue in Hebrew.

Latin and Greek

Did Christ ever use Latin or Greek? The society of first century Palestine was a multi-lingual society. Aramaic was widely used. The Hebraic Jews generally knew Hebrew as well as Aramaic. The Roman influence in the region brought the Latin language into usage. Jews who were not of Hebraic descent, who were converts or were children of converts, generally used Greek as their written language instead of Hebrew. Many important scholarly works were written in Greek; Greek scholarship competed with Latin scholarship for the attention of educated readers. Thus, the title on the Cross was written in Hebrew, Greek, and Latin (John 19:20). Though Aramaic was the most common spoken language, those who were literate used Hebrew or Latin or Greek (or more than one of these) as their written language. Christ lived in a society where Latin and Greek were used, so He had some familiarity with those languages. - The Writing of the Gospels - Which Languages Did Jesus Use?

Some believe that Jesus spoke Greek before Pilate.

“In what language did Jesus and Pilate converse? There is no mention of an interpreter. Since there is little likelihood that Pilate, a Roman, would have been able to speak either Aramaic or Hebrew, the obvious answer is that Jesus spoke Greek at his trial before Pilate.” - In which language did Pilate and Jesus likely converse during Jesus's trial?

Further information can be read here: Two Archaeologists Comment on The Passion of the Christ

And Greek was spoken in Egypt at the time of Christ.

Even in areas in Galilee where Greek culture did not dominate—like Capernaum—Greek influence was still felt. This is because the region of Galilee lay on trade routes to Damascus and elsewhere. Greek, as a language of international commerce and trade, was spoken by individuals traveling through the area.

Additionally, even though most Jews in Galilee fiercely resisted the influence of Hellenism, Greek was still spoken by select Jewish communities, especially in the south, in the areas around Jerusalem and Judea.

Greek was spoken more frequently in these areas because returning Jewish diaspora from Greek-speaking areas brought the language with them to Jerusalem. Many of them came from Alexandria, in Egypt, a region also conquered by the Greeks and still heavily influenced by Hellenism.

The same was true for other regions around the Roman Empire. As these Jews returned to their homeland, they brought with them their language—no longer Hebrew, but Greek. In fact, it’s possible that as much as 20% of the Jewish population in Jerusalem spoke Greek. - What Language Did Jesus Speak?