As a general rule, airlines will only offer itineraries that have layovers less than 24 hours. If there's no legal connection (that is, a layover under 24 hours) then most airlines won't show a ticket for sale. However, if you notice that your route includes a stopover, you may want to push out your next flight by a day or two so that you can spend more time sightseeing. Many booking websites offer roundtrip, one-way, and multi-city booking options, so if you notice a stopover on your proposed itinerary when looking at flights, choose the multi-city option and add in your next flight for a couple of days later. Sometimes (depending on airline and route) the cost will be similar to the option with the shorter layover.
Some airlines, like Icelandair and TAP Air Portugal, let you build in free or cheap stopovers of multiple days.