Hi Lara
This time of the year is a pretty slow one for openings with the holidays and the winter season seeing few moves in or out of the region. How stuck are you on finding a rental single family house? There is a complex of townhomes for rent which are nice and in a good location. Times Square Townhomes are the name and are 4 units to a whole building....side by side. They have a double garage on the bottom and then one or two levels above them...so no one above or below you just to one side. Otherwise the area has built lots of apartments in the past 7 years....although I know you mentioned you werent interested in apartments. That angle might possibly have to be considered.
The Times Square place is located on 36th Ave S just west off Columbia Rd and very near the Super Target, Kohls, SuperWalmart, and all the restaurants and shopping on 32nd Ave S. 32nd Ave S is the main shopping area in town (at least for national chain retailers).
Otherwise...most rental properties are managed by property management companies in town. Majority only do apartments, but I know Elite does homes as well. I would strongly advise to contact Elite to see how the rental home situation is in town. Their website may not have all the ones avaiable. Plus too they may steer you in another direction to check out that is not available on the web.
Here is a list of most property management companies in town. These are mostly apartments, but you might want to contact them to see about home rentals.
property management companies grand forks - Google Search
I have seen a few pop up on craiglist....
In fact I saw this ad today....
A 4 BR house in a nice circle in GF (http://grandforks.craigslist.org/apa/2103589603.html - broken link)
The overall selection for rental homes is not large so not able to be too picky.
So in final...my advice....contact as many places to see what is there as the web doest have them all.