I'm from Singapore so while I've not used the card in Singapore yet, I think maybe I can help answer some questions Monzo users might have:
In terms of public transport- we use this card called EZ-Link, which works like the oyster card. You can use monzo to pay for a temporary EZ-Link and you get a deposit back on the card at the end of it. EZ-Link can be used on buses, trains, and some taxis. Some debit cards are linked to EZ-Link but with monzo that's irrelevant in this case.
Singapore is pretty cheap in terms of food and travel: you just have to know where to go to! Avoid buying bottled drinks and food at tourist traps- supermarkets are available in any shopping centre and there are shopping malls everywhere. All buses lead to train stations, so you will never be lost. Uber is more expensive compared to normal taxis.
I would recommend getting food at hawker centres or search for local food haunts (eg Boon tong kee chicken rice, katong laksa) and in People's Park Complex you can get Michelin-star awarded food at around £1.50 ($2 SGD). Be sure to go early as it sells out quick. Avoid the tourist trap on the food street on Chinatown and try the Vietnamese food (Yummy Viet) on the same street. The food is the closest in terms of authenticity that I've found when I've compared it to what I've had in Vietnam, Thailand, Singapore, Malaysia, London, Manchester etc. The chef is Vietnamese 🏼
Singapore tends to go OTT with tourist attractions and hate boring parks so I would recommend going to Gardens by the Bay (pay to go into the domes, it's worth it. The skywalk not so much). Chinese Garden and Botanic Gardens are also very pretty places to visit. I would definitely visit the zoo, night safari, bird park. Maybe I'm biased, but I haven't found anything that tops it yet. I love feeding the Lories at the Lory Loft. There are two big aquariums and if you want to splurge there is an underwater restaurant called Cat Cora on Sentosa Island. Universal Studios is overhyped imo.
Almost everywhere accepts cards and there is less use of contactless, most places ask you to input your pin/ sign. If you go to a wet market or night market they may only take cash. ATMs are available near most transport places and shopping malls.
If you have any other questions feel free to drop me a reply! I'll only be visiting Singapore next July so I can't confirm anything for sure in terms of card usage, though.