Having travelled to Singapore many times for work (although not much recently) and obtained a licence to operate there in the past, I would suggest that you are skirting the boundaries of what you might get away with.
The first aspect is when you apply for a license, you need to specify the equipment and serial number on the license form, which in effect authorises the equipment for use and acts as a pseudo-import certificate. If you are stopped and your bags are scanned, and if they determine the radio isn’t authorised for entry, they will confiscate it. In my experience, about a quarter of the time they’ll pull a random person from the “Nothing to declare” line and scan their bags anyway. I had this happen the second time I was carrying a radio in, and I showed them my license and equipment list and they let me go on my merry way. I don’t know what would have happened if I didn’t have a license, but then I rarely seem to get stopped at customs anyway, so maybe I’d get away with, maybe not. I didn’t fancy finding out.
The second aspect is if you are transiting only in Singapore (which from your itinerary suggests maybe not on your return as you may go RTW), is that Singapore has at-gate security, so having your radio equipment etc in your carry-on may be flagged. You would leave your flight, enter the terminal area, but before you get back on board, you will be rescanned through security at the gate. I’d say it’s an efficient process, but I’d be lying, and to be honest, it’s the only thing I hate about transiting Changi Airport. I much prefer Hong Kong, which gets the transit security out the way immediately so you can relax in the lounge until they’re calling your name, but you still have to avoid the radio equipment in the hand luggage due to their import/carriage rules.
SARTS and IMDA have various pages on most of this stuff for Singapore, Google is your friend, and Harish 9V1HP is the AM and also my friend, and may be able to help you as well.
Everywhere else, as Ed mentioned, won’t care about stuff. NZ is the cruisiest of the immigration/security - unless you’re carrying organic matter - and the US will probably swab the equipment if they think it looks suspicious when you board your flight, but they won’t care on entry. Australia varies between straightforward and cranky as hell depending on the time of day. Avoid entry in the morning prior to 8am, if coming into Sydney or Melbourne (when the main curfews lift).