I realize the last post on this topic was over a month ago, but thought I would add my experience as it was in the Norwich Job Centre Plus office, also.
Firstly, I had called the number others have used via this forum. I was asked why I wanted a NIN and explained I was just married, received FLR(M) visa and was looking for work and wished to make things easy for prospective employer upon initial hire and to be taxed correctly. He set up an appointment for me in the nearest 'plus' centre (Norwich), took my details, gave me a reference number, told me what to bring, etc. Received confirmation letter via post two days later for appointment.
Appointment day - arrived 5 minutes before 15:30 appointment, greeted by friendly security person who told me where I needed to go, then greeted by friendly male receptionist who commented 'Ah, I know what that is - National Insurance Number - I recognize the letter' when seeing the appointment letter in my hand. He then asked for my passport only, told me where to sit and someone would be with me shortly. Next, a lady came over and asked for Biometrics Residency Permit card and took that with her.
Shortly after, a gentlemen came over and asked my wife and I to come over and have a seat. He was very official, but also joked around with us in a way that maintained his professionalism while making the situation a bit more pleasurable for all. He asked for the marriage certificate and then he looked at my passport (he had just made photocopies of it with the visa) and wrote a few things down while making some dry, but funny comments. He then asked about my previous trips to the UK and how long each visit was (info he wrote down from my passport). He asked when I arrived and when the wedding was. I answered and he commented jokingly 'just in time!' He also made some funny comments on how, as a male and first time married, I should put high importance on remembering the wedding date!
He asked if I had started looking for work and I replied 'a bit', as we had been busy with all the other official stuff to do after the wedding. He then replied 'you're looking for work' and wrote that down on the form as if to say 'this is what I need to put on the form for you to get a NIN'.
I looked over and confirmed personal info he wrote down on the application and signed it and that was it. He said I should have my number in about 10 days via post. He didn't ask for anything addressed to me/us at our address (council tax bills, etc) - nothing! I then asked about using the job centre to help find work he said basically 'don't bother, everything we have is also on the internet - save yourself a trip and do the same thing at home with a cup of tea and a computer'. He asked what type of work I did and when I told him he said 'you won't have any problems'.
We spoke more and answered more questions for this than the PEO FLR(M) appointment we had! But all said and done, it was quite easy and painless and made a bit more pleasurable by a the person taking the application info.