Why don't churches display the papal flag, either inside or outside on a flag pole? Shouldn't we show our allegiance to the Pope?
--Furious in Flagstaff
There are no rubrics which mandate flying or posting a papal flag in church. Like most flags, they symbolize a country/national identity. The papal flag is actually the symbol of the Vatican City state/country. The pope has a special coat of arms which symbolizes his "theme" or "motto." These are two different "logos" for two different purposes. In 1825, when much of what we call today as Italy comprised the Papal States, this flag was introduced to symbolize the country/state. In 1929, when the nation of Vatican City was established, this flag was kept to represent the politicital entity, the government, rather than the Catholic church itself. Its use is governed by the same principles as the flag of any other nation. For more information - click here.
[Special thanks for this answer go to Thomas Wlebers, "The American Flag in Church?" Rite Magazine, July 2002 Chicago: Liturgy Training Publications, pp. 4-7.]