It really depends on the non-sql solution... Most of which do have indexed keys for retrieving data at the very least (key-value storage), and use Map-Reduce in order to perform basic queries, or you setup your own indexed records.

MongoDB specifically allows for additional indexes to be used as part of general queries, or the aggregation methods available. MongoDB even has some nice built in features for querying against geolocation data. They do have some limitations, but in general it works quite well.

Your analogy comparing non-sql to a trashcan is a bad comparison... a better comparison would be to a set of file cabinets.