There are two options of booking this: A) as a single itinerary with a stopover, or, B) as two independent tickets.
Option A) is dependent on the stopover rules of a specific ticket. These can be complicated, e.g.: none, one per direction, unlimited, various fees, etc. This doesn't lend itself easily to search engine integration and since demand for this is rather low, no engine I know has bothered trying to integrate it.
Even most airlines keep that somewhat opaque: Stopovers can only be booked as a multi-city ticket and a ticket from A to B to C could be issued as a "A->C with a stopover in B" or as two separate fares "A->B" + "A->C". This being said: Some airlines (e.g. https://www.cntraveler.com/story/airlines-with-free-stopovers) do actively advertise free stopovers in their hub.
Your best shot here is to do one of the following
- Look for BUD->SIN, identify the top 5 choices that meet your criteria for attractiveness, detour, tolerable pricing, etc and manually check the stopover rules for each fare
- Look for "extra long layovers" with ITA matrix. While MINCONNECT 36:00 does indeed come up empty, MICONNECT 24:00 or MINCONNECT 30:00 provides a good list of options. This can also be a great starting point for method 1) above.
An extra long layover may have the advantage that the airline may hold on to your checked luggage. For a stopover you will always have to collect and recheck bags.
Option B (two separate tickets) is a non-starter for a search engine. There are several hundred potential stopover airports each with dozens of flight options to/from BUD and SIN. The sheer number of possible combinations will be prohibitively large.
So you would have to constrain the search somehow. Both BUD and SIN serve 100+ destinations. If you want to minimize the amount of flying, you could look for destinations that are served by both airports and geographically desirable (by your preferences, visa, etc.). A good tool for something like this is flightconnections.com.