The pope lives in the Papal Apartments in the Apostolic Palace in Vatican City.
For 68 years, 1309-1378, the papacy was headquartered in Avignon, France, rather than Rome.
There are no popes in Singapore, only Pope Francis in Rome.
the vatican
After Pope Benedict X died, Bertrand de Got of France was elected. He was to never to step foot on Italian soil, so the following six French Popes chose to stay in France as well.
The fortunes of medieval Rome rested on being the seat of the papacy. From 1309 to1377 seven popes resided in Avignon in France, instead of Rome (the Avignon Papacy). With the absence of the popes, the population of Rome decreased and the city was neglected. When the popes returned to Rome they undertook ambitious programmes of reconstruction and urban regeneration.
The popes have always been the Bishops of Rome, and so they have mostly lived in Rome, throughout the history of the Church. There have been times when the popes did not stay in Rome much, or did not stay there at all. There was a time from 1257 to 1309 whose popes are called the "wandering popes" because the political situation in Rome was such that they were unsafe there. There was also a time from 1309 to 1377 when the popes lived in Avignon. Aside from these times, there were papal palaces in different places where the popes would stay when they had some reason to be there. There is a link below to the an article on the history of the papacy.
There have been 266 popes, the vast majority ruling from Rome, so you really have to narrow this down a bit.
rome
The popes of the Catholic Church have always been the bishops of Rome. That being the case, they nearly all lived in Rome, in a papal palace. There were other papal palaces in other cities, and there were times when the popes did not live in Rome at all. There was a period from 1257 to 1309, when the popes were said to have been wandering, because they rarely went to Rome because of political situations in the city. There was also a period from 1309 to 1377 when they lived in Avignon. But mostly, they lived in Rome.
The papacy's stay in Avignon lasted from 1309 to 1376. Therefore, the popes were out of Rome for 67 years.
In 1309, Pope Clement V moved the Roman Catholic Papacy to Avignon in Provence France. From 1309 until 1377, seven Popes reigned in Avignon before the Schism between the Roman and Avignon churches, which led to the creation of rival popes in both places. After that three Antipopes reigned in Avignon until 1423, when the Papacy finally returned to Rome.
It is a beautiful church in the Vatican in Rome. It holds the popes, art, and mosaics.