Have no first hand experience of Oman but found this for you and hope it helps. Sorry it's so long.
Here is my humble - and very general - guide, based on my own experience of travel in the region, from Saudi Arabia to Syria to Turkey ... and elsewhere. Having been a solo woman traveller myself, I understand the anxieties expressed by many on this board. Fortunately, however, I also understand that most of these fears are unfounded.
Bear in mind that there is much of the ME that I have never been to, and that the advice below is based only on my personal experiences. Therefore, please feel free to add to, contradict, or debate anything I have said. All in the name of well-informed travel!
Will I be safe?
Nowhere is 100% safe. However, the ME in general is not a dangerous place for the traveller, and, while petty crime can be an issue, random violent crime against women seems to be very rare - probably much rarer than in most Western cities.
Will I get hassled?
Quite possibly. While you need have no particular worries about personal security, sexual harrassment can be a problem. Reports of hassle vary widely, but it seems to be at its worst in parts of Egypt and Turkey, much more low-key in Jordan. Everywhere you go, you can expect to be the object of great attention and curiosity, and may well have to deal with a few incidences of being followed or prepositioned. However, it should get no worse than that.
But what if it does?
As anywhere in the world, the best response to harrassment is no response. Remember that it is most certainly NOT acceptable for a man to harass a woman anywhere in the ME. Therefore, if you do feel that a situation is getting out of hand, by all means elicit the support of those around you (this is another area where modest dress will help you win friends and influence people!) No need to shout or scream - just calmly point out the offender to some bystanders, and it is very likely they will put him in his place. In societies where notions of 'shame' and 'honour' are paramount, public disgrace can be a potent weapon: use it!
How can I avoid hassle?
The first priority is to dress reasonably (see below). However, even if you're wearing a Saudi style abaya, it won't help much unless you also behave appropriately. On the street, do as local women do and try not to make any eye contact with men. Sunglasses can be a great help here. In your interactions with men (and most of the people you deal with will be men) be polite but reserved. Never volunteer the information that you are travelling alone: if asked, say you are with friends or a group, though you may not be believed.
Be especially careful not to smile or giggle too much - what seems to you to be an 'innocent' smile might well be misconstrued as a sexual invitation. Try to keep conversations 'business-like', and do NOT allow yourself to be steered towards matters of a more personal nature. Never EVER allow a man to touch you: while Arabs may be quite tactile with members of the same sex, it is absolutely not on for a man to touch an unrelated woman, even 'accidentally'. If a man touches you, however 'innocently', let him know in no uncertain terms that you consider his behaviour inappropriate, and extricate yourself from the situation asap.
This need for caution is a pity, because in countries where the public sphere tends to be male-dominated, men may be your only point of contact with the local culture. It needs to be pointed out, of course, that by no stretch of the imagination do all - or even most - local men have 'dishonourable intentions' towards you. Those dealing with tourists, should, in addition, be aware that just because a woman may be willing to chat to a man does not (neccessarily) mean she is also willing to sleep with him! Also, throughout the ME, people are naturally friendly and talkative, loving nothing better than to pass the time chatting with foreigners. In general, however, it pays for solo women to err on the side of caution.
What should I wear?
The No. 1 question from women travellers on this board! Unless you're going to Iran or Saudi Arabia, you are in fact free to wear whatever you wish - and many travellers do just that. In touristy parts of Egypt, the UAE, and Turkey, you will see plenty of foreign women dressed in skimpy shorts and sleevless tops, and, in many places you will also see some local ladies dressed less than 'modestly'. Outside of the above mentioned countries, nobody is going to arrest you for showing too much flesh, but you might like to consider the impression you are making. While modest dress will not in itself guarantee you won't be hassled, it will at least reduce the chances that you will be seen as a 'foreign floozy' by all and sundry.
So, what's a gal to wear? No need for the 'hippy look': you will notice how many Arab women manage to dress both modestly and stylishly at the same time. Local women pay a lot of attention to their appearance: often you will see women so fashionably dressed that it barely registers that they are in fact covered from head to toe! T-shirts or blouses which extend at least to the elbow are fine, as are loose skirts or trousers (preferably full-length but at least knee-length). If your trousers are a little on the tight side, take a tip from local women and wear a long blouse or jacket which covers the behind. High heels are acceptable, but not terribly practical!
Do I need to cover my head?
All women are expected to cover their heads when entering mosques. However, with the exception (again!) of Iran and Saudi Arabia, foreign women - and local women, for that matter - are under no obligation whatsoever to wear a headscarf. In fact, unless you are a Muslim and are used to the headscarf, covering your head is likely to make you the object of more, not less, attention. While some people might appreciate your desire to 'show respect for local customs' the majority are more likely to see the gesture as rather eccentric, particularly as many local Muslim women have never worn a headscarf in their lives!
I've got long blonde hair. Should I dye it/ cover it/ chop it off?
Another TTME classic! In parts of the ME - such as Syria or Turkey - it's not unusual to come across locals with fair hair and blue eyes. Also, many Middle Eastern women dye their hair all shades of blonde or red. While it's true that blondes stand out from a crowd, your hair colour is not likely to be an issue in itself: foeign women (such as myself!) who have dark hair and dress modestly are still instantly recognised as foreigners. Whatever your hair colour or style of dress, as a solo woman traveller you are sure to attract a certain amount of attention.
I plan to travel alone. What will people think of this?
In ultra-conservative countries like Saudi Arabia, no respectable woman would make even the most routine trip without an escort. Things are not so extreme in the rest of the ME, but it is still unusual for women to undertake even short journeys unaccompanied. Arabs tend to be very sociable people who find it hard to understand why anyone - particularly a woman - would actually choose to travel alone. It's not that they disapprove of what you're doing, it's just that they find it distinctly odd! Rather than tut-tut at you, they are much more likely to pity you, alone in the big bad world without a man to protect you! Which brings us on to...
If people ask if I'm married (I'm not) what should I say?
"Are you married?" is invariably the first question any woman over the age of 18 gets asked in the Middle East! Although things are changing, in most parts of the Arab world, all but the most sophisticated women marry early - usually in their early 20s, but often in their late teens. While many Middle Easterners know that social norms are different in the West, they still find it very strange to meet a woman over the age of 25 or so who seems happy to remain single. Once again, it's not that they disapprove of you, more that they pity you!
Rather than deal with all this bafflement and unwanted pity, many women invent a hubby who is "resting back at the hotel"! Which, of course, will lead onto the inevitable follow-up questions: "Do you have children? No? Why not?" (try responding to their concern with "Insh'allah" "God willing") Some women go so far as to get a cheap gold band or even produce photos of their 'husband' to add credence to their claims of being lawfully wed.
To lie or not to lie? It all comes down to a personal choice. As well as fobbing off unwanted pity, some women say that a fictitious hubby scares off would-be suitors, though others say that men in more touristy areas are 'wise' to the scam and will not take your 'married' claims seriously.
I'm travelling with a male friend. Should I lie and say he's my husband or brother?
Again, entirely up to you. Many more traditional Middle Easterners aren't familiar with the concept of 'platonic friendships' between men and women, nor do they have 'boyfriends' or 'girlfriends' in anything approaching the Western sense. If you are dealing with people like this, you might find it more diplomatic to lie about your relationship with your travelling companion: it's not that the truth will get you into trouble, just that a little white lie might make things less complicated. However, younger, more sophisticated Middle Easterners will have no problem with the idea of your travelling with an unrelated man, and there is no need to invent any 'wedding bells' in this company!
Happy travels!