I recently needed to use the plural of lingua franca. I thought that I might show off by writing linguae francae but a routine check in the COED before I committed myself showed that the only accepted plural is lingua francas. The reason is that the expression is not Latin in origin but Italian, meaning Frankish tongue.
This use of Frankish dates back to the Crusades and derivative forms are found around the Middle East to mean European/Christian in general.
Because the Frankish kingdom dominated Western Europe for centuries, terms derived from "Frank" were used by many in Eastern Europe, the Middle East, and beyond as a synonym for Roman Christians (e.g., al-Faranj in Arabic, farangi in Persian, Frenk in Turkish, Feringhi in Hindustani, and Frangos in Greek). See also Thai ฝรั่ง Farang. During the crusades, which were at first led mostly by nobles from northern France who claimed descent from Charlemagne, both Muslims and Christians used these terms as ethnonyms to describe the Crusaders. Another term with similar use was "Latins" (cf. the Latin Empire). This usage is often followed by modern historians, who call Western Europeans in the eastern Mediterranean "Franks" or "Latins" regardless of their country of origin. Compare with Rhomaios, Rûmi ("Roman"), used for Orthodox Christians. Catholics on various islands in Greece are still referred to as Φράγκοι, "Frangoi" (Franks). Examples include the naming of a Catholic from the island of Syros as "Frangosyrianos" (Φραγκοσυριανός). The term Frangistan was used by Muslims to refer to the land where the Crusaders came from, i.e. Christian Europe.
The OED offers no specific recommendation for a plural but its only two plural examples make the Italian lingue franche:
1971 J. Spencer Eng. Lang. W. Afr. 31 A very complex infrastructure of scores of vernacular languages as well as a number of regional lingue franche. 1974 R. A. Hall External Hist. Romance Lang. 21 The distribution of the Romance languages is best treated under four heads:‥(3) use as lingue franche.
This seems unnecessarily pedantic for the non-linguistic context in which I am writing, so I have decided to stick with lingua francas.
Wikipedia has a List of lingua francas.