Have a read of the thread below. Also worth trying another sim to see if it is a sim issue.
/t5/Nokia-with-Windows-Phone/Lumia-800-Battery-wif
If you can't or won't do that then you should return the phone to the retailer for an exchange or contact Nokia care.
And how should I do that? Tape the sim or sth?! 
If I shake my phone heavily, so that my sim card has to move a bit, the phone doesn't reboot. It makes me hard to believe it's the sim card then.
Dear Nokia support, are there any other reported solutions for this issue? I am not the only one if you look here or into other communities and perhaps there were similar issues in the beta testing? Maybe a hard reset is the solution?
If not, what should I do? Write to the the support? Go to my shop and demand a new sim card or a new phone?
If it's really the sim, then please inform your partners, that the lumia 800 is sensitive about sim cards - so it's not a problem for me to explain the shop why I need a new one.
Because if I go now, they are going to say something like "yeah, look: the sim works, it's brand new and extra for your phone - you can call sb, internet works, sms works. The sim card is ok."
First of all it would be enough if you say, that you realise and investigate this issue. I can live with random reboots a few weeks until you worked out some solution possibilities, but it's not a durable solution for 2 years.
