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Great question, I'd also love to hear some opinion of locals about which dish to look out for.
All of the dishes I've tried so far are from every Asian country except Singapore.. skimming through Wiki pages I couldn't find any dish specific to Singapore either.
After trying food courts for a couple days I decided to check grocery stores since I got a very little amount of vegetables with all the dishes making my daily food input much less healthy than I am used to. In those grocery stores I noticed that the majority of products were from western countries or from Australia (even a milk package saying '100% Australian!' ....someone really seems to be making sure he's not far from home, haha^^) . Maybe I was checking the wrong stores?

I guess that's what you can expect from one of the most westernised Asian countries but surely there's still a dish favoured by the majority of locals?

Since Singapore shares a culture and cuisine with surrounding countries (especially Malaysia) and has absorbed culinary influences from many others, the uniqueness of Singapore food lies not in spacific dishes, but in the sheer variety.
However, there are some dishes unique to Singapore: Fish head curry (which was invented by South Indian immigarnts in Singapore) and Chilli Crab come to mind. There's also Singapore Laksa and Fried Rice, which are markedly different from their namesakes in nearby places.
And, although it is also present in Malaysia, Peranakan cuisine is the nearest Singapore has to it's own style of cooking.