My wife and I have this plan: retire, sell our home, and travel abroad indefinitely. We'll have no permanent physical residence. We are US citizens, and our legal residence and domicile will be in Florida, where we'll have driver's licenses and voter registration. We'll have an address at a mail-handling service. We'll also use the home address of a relative in Florida for some things that prohibit using a mail-handling address.
We'll stay in each country or visa area (eg: Schengen) for several months at a time on tourist visas, rarely returning to the US. I assumed that showing sufficient funds, proof of lodging in the host country, and an onward or return ticket would allow us entry into the various popular tourist destinations around the world.
However, this question has me wondering, because the OP is a US citizen and was denied entry to Canada partly because the OP "couldn't provide proof of permanent residency in the USA", such as utility bills. Presumably Canada was worried that the OP would not leave after entering. How big a deal for US citizens is a lack of permanent residence when attempting to enter foreign countries (not just Canada) on a tourist visa, intending to stay for several months? And even if we maintain a permanent residence in the US, would other countries have a problem if we almost never go back to our home country?
I'd like general answers, but if the question is too broad, then assume we are entering the Schengen area.