I have been to Singapore for both personal and business trips but never to Perth.
Ok, there are lots to do in Singapore. First of all, you should know it is the "terminus" of the Japanese 25th Army's aggressive drive down the Malay Peninsula from the Thai border on down. Singapore is a historical battle site and the city state has done a LOT to attract military history buffs to visit the island. Furthermore, Singapore have quite a number of hobby stores so that serves as a bonus.
For hobby stores, I could write pages upon pages on all the stores I visited there. On the other hand Chris Meddings (Inside The Armour) currently lives in Singapore and maintains a VERY helpful blog on all the hobby stores he's visited:
The page linked above features M Workshop, if you are going to miss all other hobby stores, do not miss this one. Lots of Missing-Lynx regulars from Singapore have their works featured in the storefront. Not to mention a store where the owners like to talk "shop" and you will often see people working on kits.
Chris have other stores featured in the blog.
Historical site wise, you should visit the Battle Box, which is a 20 minutes walk from M Workshop, situated on Fort Canning hill. The Battle Box is where General Percival held his last war council before deciding to surrender to the Japanese. There is a modest admission fee to the complex, but you get to walk around in the very same underground bunker and listen to pretty convincing animatronic wax figures discussing the Battle of Malaya (and Singapore). For a WWII buff, this is definitely worth a visit.
You also may end up on the island of Sentosa since it is a tourist trap. On the island is a Universal Studios amusement park, several beaches (featuring almost NO waves, by California standard) and most importantly Fort Siloso. Fort Siloso is an artillery museum with big guns from every age of Singapore's history. They have 8-inch barrels from British cruisers and even 25 pounders used by the Singapore Army a few decades ago. The fort also has a walk-through bunker complex again with wax dummies and animatronics showing you life in the fort around the WWII timeframe.
Personally, I would pay the Gardens By The Bay a visit around the Marina Sands mall/hotel/casino. The gardens are the largest greenhouses in the world featuring climates and plants from different parts of the world. These greenhouses opened recently (in June) and are a big attraction.
The gardens are situated right next to the Marina Sands mall and you can eat/shop afterwards.
Also, Singapore is a foodie's heaven with restaurants featuring cuisine from all over the world. I personally prefer the local hawker center fare as it is cheap and extremely clean (their government keeps a tight rein on food vendor cleanliness.) Plus, they are like nothing you have tasted in the western hemisphere. Skip the "Singapore Sling" in the historical Raffles Hotel, that is a tourist trap.
I do not live in Singapore.
But that little island is one of my favorite destinations on Earth, next to Yosemite in California.