Hi,
As far as I know, it is certain that foreigners can rent an apartment in Vietnam without any difficulties. Coz if there is any problems that you are unclear about or unable to tackle on your own, it must be your host, who will take charge of all especially the procedures concerrned with the local gorvement.
One my friend, an America lady used to rent an apartment around the Old Quarters(hm... on Ma May street). Everything went on smoothly then, except for the negotiation on the rent with the landlord. And I am afraid that it is probally one of the crazy stuff you should also pay your attention to^_^.
Ok, take your time in Hanoi and enjoy it. Good luck!

You can rent an apartment as far as I know, but to rent to a foreigner the landlord is required to be licensed to do this although many are not. They should also supply you with a 'red' invoice although again many do not or require 2 contracts - one for the actual amount and one for the amount that is paid to the tax authorities

In other words; it's quite impossible to move to smaller cities? I don't think there is other foreigners where I am than me (other than tourists of course). And I doubt anyone would be licensed to rent to foreigners here.

Where are you?

In Ninh Binh right now. Moving to an even more odd place soon ;)