Now, as a quick jaunt through your nearest major city will most likely tell you, this isn't the case in the real world. During that jaunt you'll likely hear at least two different languages and see signs in a third and fourth. Of course, through out history numerous different languages have been created or used as a common language between speakers of two different mother tongues, and that's what this article is going to be about. Hopefully this will help you add a touch of verisimilitude to your setting.
The simplest definition for pidgins is that they are a language formed on an ad-hoc basis to enable communication between two or more speakers that don't share a common tongue. Pidgins are generally lacking in any grammatical rules and are built from sounds, gestures, words, and body language, often time they use verb reduplication to express plurals and superlatives (For example: "Unlove Big Brother ungood ungood plus," to mean "Not loving Big Brother is a horrible crime.")
This view of Common generally precludes anyone from being a native speaker of it, or particularly fluent for that matter, and having a philosophical debate with the barkeep should be extremely difficult, if not impossible, using it.
Chinese Pidgin English
Exemplified by the phrase "Me love you long time" from Full Metal Jacket, this is the origin of the word pidgin, featuring a vocabulary of approximately 700 English words and a grammar determined by the position of the word in the sentence. This pidgin died out in the 19th century as English began to be taught in Chinese schools.
Straddling the line between pidgin and creole, it originated in the Pacific Northwest as a trading tongue between the various tribes in the region. As European traders reached the coast it gained words from French and English. It had a simple vocabulary, with the number of words numbering in the hundreds, and unlike many pidgins it had a grammatical system, though a simple one, making it easy to learn.
The origin of the term Lingua Franca, Sabir was originally based heavily off of Italian, but as it was used as a trading tongue by peoples throughout the Mediterranean it picked up vocabulary from the entire region, to the point where a Frenchman and an Arab could speak to each other in it and legitimately believe that they were actually speaking each others language.
Creoles are the bastard tongues of the world, often times taking the grammatical structure of one language and using the vocabulary of other languages. The biggest difference between creoles and pidgins is that they possess features of natural languages absent in pidgins, and are often times the first language of a minority or otherwise dispossessed group in a region. The simplest reason for a creole language is that it is the natural outgrowth of a pidgin language, taught to the children of a group that speaks a pidgin tongue as the children's primary language, at which point it develops structures and grammatical rules not present in the original pidgin.
The spoken language of the island-nation of Jamaica, this creole is heavily influenced by English, with additional influences from West African sources. Differing significantly from Standard English, native speakers of both languages can understand each other to a degree, however the differences in pronunciation, grammar, and vocabulary would make them mutual unintelligible if it wasn't for the fact that constant trade between them and other parts of the English speaking world allows a familiarity with each other.
The language spoken in Haiti, based off of French with influence from various African languages, Arabic, Arawak, Spanish, Taino, and English, it is one of Haiti's two offical languages, the other being standard French. Only officially recognized as an official language in the 1960's it has only recently become accepted to use Creole as a literary language.
Defined as a situation where two different dialects or languages are used by a single community, with one of them being viewed as a low (or vulgar or vernacular) tongue and the other being viewed as high or literary speech. The low tongue is generally used in common everyday settings, such as at the market, around the house, or when talking to friends and family, and the rules of the language and pronunciation are less stringent than the high tongue, it might be a creole or a dialect of the language that differs noticeably from the high speech. The high speech on the other hand will be used in more formal situations, such as literature, higher education, or politics and is rarely used to converse with someone.
It is usually not possible to acquire the high tongue though anything other than formal education, due to the fact that there are generally not enough speakers of it to just pick it up, and those people who do speak it will generally decry your butchering of their beautiful tongue when you try.
Also known as Jive or by it's more proper name of African American Vernacular English, Ebonics is predominately spoken by lower class African Americans. Viewed by the majority of society to be a socially unacceptable and down right degenerate English dialect that only proves how the people that speak it are lazy, criminal, and uneducated. Whether this is a true form of diglossia or not is up to actual linguists to decide, I'm just providing it as an example that most English speakers who've been exposed to American media can understand.
The primary language of 22 (or 24 counting Palestine and Western Sahara) countries in the Middle East and North Africa, Arabic is a very good example of diglossia. Public speeches, books, laws, and satellite news are all delivered in Modern Standard Arabic (MSA), a language that differs little from the Classical Arabic that the Quran was written in. However, no one actually speaks MSA, by which I don't mean that no one speaks in MSA but that no one speaks it as their primary, day to day language. For that they use the vernacular dialects, be it Cairene, Iraqi, Sudanese or Maghebi, which are generally unintelligible to speakers of a different form of Arabic, and are almost entirely without a written form.
Classical and Vulgar Latin
Classical Latin was spoken in the 1st century BC, after that the Vulgate predominated as the spoken language. Despite this, Classical Latin was still viewed as the only truly proper way to write anything important until the 17th century, turning many of the Romance languages into the low languages of the time.
High and Low German
Low German was the lingua franca throughout the Hanseatic League for nearly two centuries, but it began its decline with Martin Luther's translation of the Bible into High German. Other factors affecting its loss in prominence, and the low view that many native German speakers hold of the language is that the Hanseatic League lost its importance in the region as new trade routes were opened to the Orient and the Americas during the 16th century, and the mass education of standard High German during the 18th and 19th centuries.
Translated as "Language of the Franks" this can be used to describe any language, be it pidgin, creole, or full language, used by people to communicate when dealing with speakers of another language. However, for the sake of needing something to describe this section, I'm going to say that it is a vernacular language used by people who generally don't speak it as a native language, but share no other language in common, to communicate with each other.
Due to the fact that Alexander the Great conquered much of the western world in the 300 BC, Koine (the Greek word for Common) was firmly established as a language that many different people of various cultures spoke, and was used as a language of trade and communication well into the time of the Roman Empire. Koine is also known as Biblical Greek due to the fact that the New Testament was originally written in it.
Chinese served as the primary lingua franca throughout the Far East up until the 20th century, at which point it was largely replaced by English. China's central position in the region served as one of the motivating factors in its use, as well as the fact that the countries neighboring China were consumers of the Chinese culture.
The language of scholars throughout Europe up until the 19th, Latin was brought by the conquering Roman Legions as far west as Hadrian's Wall in Britain. It maintained its hold largely thanks to the power of the Catholic Church, which was the primary source of education for many until the Protestant Reformation.
The primary language of European aristocrats and diplomats from the 17th century through the 20th. Many international institutions still use it as one of their official languages. Due to colonial influences, it is still used as a lingua franca in many parts of Africa and some parts of the Middle East.
There was a time when the sun never set on the British Empire, English began its rise into its current prominent role between the two world wars, and reached its current position of world wide lingua franca after the Second World War with America's rise to a world power and the fact that much of mainland Europe was in no position to argue with it. It is now cemented in its position due to being the language of Maritime and Aviation.
Pidgin, Creole, or Lingua Franca?
The choice of what exactly Common is in your setting is one of flavor. Pidgins are generally more likely to arise when the cultures involved are not trying to actively destroy each other, or when the speakers are only in contact with each other for very short times and neither wants to go through the effort of learning more than a few phrases in the other's language.
Creoles are more likely to develop when one group of people are obviously dominate over the other, as evinced by the fact that most modern creoles were created approximately 500 years ago during the Colonial Age.
Lingua Francas are most likely to be used when there is a significant cultural reason to such as it being the language of the Church, it's the language of the nearest large power, or the language of the Great Fallen Empire of the Silver Age.