score:3
I did not really get why you wanted to do that (I suppose you want to visit London on a transit visa), but to answer your question, most flight search websites have the multi-city search feature (you search for X-London and London-Singapore) or sometimes you can pick your transit location.
Momondo lets you search for your pair origin-destination, and in the left panel of the search results, you can click on one of the suggested transit airports, or uncheck those you don't like.
I tried expedia and kayak and they offer multi-city search. Skyscanner, on the other hand, does not seem to offer any of these.
momondo does not seem to offer either transit airport filtering nor multi-city searches. However, the Kayak app offers both airport filtering (including transit airport filtering) and multi-city search.
Upvote:1
Google Flights has functionality where you can filter connecting city. When filtering it will still show you direct flights between your two cities as well as those with a connection in the selected city.
To maximize your time in London, you may want to use a multi-city search with arrival and depature in London on the same day or across two days. I'd try this because flight search engines generally find the shortest connection, while it sounds like you're interested in the longest one. I'm not sure though, if booking a multi-city flight is handled any differently immigration-wise than a transit flight.