The short answer, before your city's
On your own land, there is generally nothing to apply for. Driveway, side yard, back yard: in the vast majority of the cities we serve, no permit, no certificate, no step to take before the box goes down. A few cities add a placement condition rather than a permit — keep a set distance from the street line, stay out of the front yard, leave a clearance, provide a screen — and they publish it in black and white in their zoning by-law.
On the street, it is the opposite. The roadway, the alley and the sidewalk belong to the municipality: occupying them almost always calls for an authorization, under one name or another — public-domain occupancy permit, road permission, certificate of authorization. It often comes with fees, which vary from city to city, and with a delay: several cities require the application several business days before delivery. And in a handful of the cities we serve, the answer is simply no: leaving an object there is an offence. In some, no procedure exists to get around it; in others, the only door is an exceptional permission, granted case by case when the land is too small.
Then comes the trap: the dumpster permit and the work permit are two different things. A city can require absolutely nothing for the box in your driveway, and still require a permit for the renovation you are doing. That is the section right after the checker, and it is the one that saves the bad surprises.
Why so much variation from one city to the next? Because there is no common rule. Every municipality writes its own by-law on the occupation of the public domain and its own list of certificates of authorization; in Montreal the question is settled borough by borough, each with its own schedule. Some cities name the dumpster in so many words, others never use the word and settle it at the counter. And several go the other way entirely: they require a dumpster on a permitted job site, as a condition of the building permit. Hence this page — someone had to go read, city by city, what each one actually says.