I lean towards shorter/simpler trips with kids.  They'll be having all kinds of new experiences/sights to process and it can get a bit overwhelming and confusing to remember what they saw where. 

It's important to identify specific attractions in each location (and maybe the kids can help research or pick their favorites from your selection).  Though you don't have to plan out each day, knowing what the options are can give you an idea of how much time you'll need in any location, and that can help you decide if you want to add another location. 

Another possible reason for shorter trips is because kids have a social life back home.  They might miss out on important events or simply miss their friends, their routines.  You could discuss this with them to see how they feel. Teens seem to be more reluctant to leave their friends but not all teens, so you have to assess and reassess as they get older.

Due to the long nature and high cost of most flights from Australia, I'd lean towards visiting a couple of places on each trip that are in close proximity to one another.

I am not an expert on Singapore. I would ask on the forum for that particular destination what amount of time they think is sufficient to see the basics. I would then ask them if there's any place nearby that would pair well with Singapore.

I agree with points made by both of the previous posters. I agree to relax and really take the time to immerse yourself in one particular destination. On the other hand, kids don't have long attention spans. They also don't like seeing similar attractions in succession. Our kids got tired of visiting all the different thermal areas in Yellowstone and staged a revolt when we went through the final thermal area.

You are in a prime window for traveling. Your kids are old enough to handle most attractions. Once they hit the mid-teens, their parents aren't fun anymore and they have jobs that don't allow time for long vacations.

Depends on who you are. If you're a thrill seeker, then multiple small trips can give you plenty of thrills. If you're someone who loves preparing for a trip, planning and having a goal, then one big trip will probably give you more satisfaction. Different people have different preferences. But, it's worth having small trips. Multiple small trips will keep your excitement levels higher. And, if you're an introvert, and you want to get used to being in new places, and talk to new people, then multiple small trips is a must.

A $5k budget is not going to get you to Europe from Australia.

South-East Asia would be your best bet. You could do Singapore on its own or you could use it as a stop on the way to Malaysia, Borneo, Vietnam, Thailand and I’m sure there are others.

When my kids were 10 and 11 we did a 6 week trip to Malaysia and Singapore. Our itinerary was:

Singapore- 4 days

Kuching - Borneo fantastic, so much to do. 8 days

Flew to Penang 7 days split between Hard Rock and an Airbnb

Train to Ipoh - another great place, my favourite part of the trip.

Train to KL. 5 nights

back to Singapore, stayed for several days.

Pre-kids we also went to Vietnam and Sabah, both places we’d like to take the kids to. I’ve been looking at Intrepid Family tours (we used Intrepid for our Sabah tour) but I ours are now 14 & 16 and if we don’t go to Europe next year we will miss our chance.