Oman is a great country and a fabulous place to live and own a motorbike. It is a pretty well structured and organised country, pretty much along Western lines really, you need to do everything properly, licence, insurance, not drinking and driving, wearing a helemt, just normal civilised stuff really. Except for chicken chasers (50cc mopeds) everything has to be properly registered else the police will feel your collar
You may want to have a bit of an investigation into what status you have here - your wife will get a residents visa and card, and you definately need both those to leap through all the admin hoops. You need your residents card when you buy insurance, register your car/bike etc although I guess your wife could do it all in her name and just let you hoon around on the bike all day
When my girlfriend used to visit me here, on 3 month visas, she would use my Land Rover no worries at all. The vehicle is what is insured not the driver, so if the vehicle is insured, you are good to go.
There are plenty of biking options out here, some easier to source than others. There are tons of Harley Davidsons around, and a lot of guys go cruising on a Friday morning. There are sports bikes galore, mainly ridden very fast by lunatic locals and there are a lot of CRF450Xs which are perfect for hitting the Wahiba Sand Dunes. These tend to get trailered to where ever you want to start, and whizzed round for a couple of hours.
If you want to do all your exploring by bike only, KLR650s come up for sale quite regularly and they are the perfect dual sport for out here, although a bit of a handful in the sand and a million miles behind the CRF in offroad ability
The distances in Oman are pretty big (well they are for someone from the UK) and I used to find I got pretty hacked off with having to ride for a couple of hundred K's to get somewhere new on a chugger like a KLR650, which I guess is why a lot of people prefer to take a 4WD and a trailer and a CRF and a ton of beer and head off somewhere interesting.
Its a great country, I spent an hour before sunset this evening whacking around the dunes on my CRF and it felt good.
Oman is a friendly happy country and a great place to be a keen biker. But one word of caution, the standard of driving is outrageously bad and there are over 1000 road deaths a year, with a population of under 3 million folks. In the UK, it is 2500 road deaths a year with a population of well over 60 million. I can't do the maths, erm, suffice it to say, you gotta take it very very cautiously if you ride on the road a lot. Which is another reason for having a big 4x4, a trailer and a CRF
But whichever way you go, it's all good