A couple of good answers about the general practices for US cities in this previous thread.

In 1929, New Yorkers put their household garbage in waste cans very much like today, though not yet the rolling carts that could be emptied mechanically. Those living in large apartment buildings could put garbage down a chute to be burned in the building's incinerator. It's worth noting, though, that the volume of refuse was quite a bit less because food and other household goods didn't come in such substantial protective packaging as today. On the other hand, apartment buildings produced large volumes of coal ash from their furnaces, which today burn natural gas instead. In the 19th century city, much household rubbish had enough secondary value (to feed chickens or pigs, to burn as fuel, to recycle) that private scavenger services were willing to collect and sort it, but that situation changed rapidly after World War I.

Most of New York's garbage was dumped at sea until the practice was outlawed in 1934. Ash could be used for landfill projects such as LaGuardia Airport, but much ash and other garbage was dumped in outer borough landfills, including a large one (Flushing Meadows) in Queens that was capped and used for the 1939/1940 and 1964/1965 World's Fairs.