The only official language of the Falkland Islands is English, and this is spoken by almost everyone on a day to day basis. Spanish is spoken by nearly 10% of the population, a significant minority. Most of the Spanish speakers are immigrants, foreign workers, and expats, predominantly from Chile and Argentina.
Due to the isolation of the islands, the small population retains its own accent/dialect, despite a large English immigration in recent years. In rural areas (i.e. anywhere outside Port Stanley), known as the "camp" (from the Spanish "campo" meaning "field"), the Falkland accent tends to be stronger. The accent has resemblances to both Australia-NZ English, West Country, that of Norfolk and another one of Suffolk, both in England.
Several languages have been used historically in the Falkland Islands.
French - the French were the first to colonise the islands, and their settlement at Port Louis would have used French. (The islands' French name, Iles Malouines, stems from Saint-Malo, and there were many Breton sailors round these parts, so Breton may have also been used, although there is no evidence of this)
Spanish - was used in early Falklands history, and also during the brief Argentine occupation of 1982.
The Falkland Islands Fox was previously hypothesized to represent a possible pre-European landing on the Falklands, but this has since been refuted. This language has left no trace on the Falklands, and would not have been written at this time.