edit:

I'd say on or with regard to

The OP doesn't really make much sense to me. I assume what's meant is: Let's see how this service compares with traditional institutions regarding ordinary matters.

I'm not sure how a service can rival or compare with an institution, though. And what are "ordinary matters"? This all seems like apples and oranges.

agreee with # 3, in

compare would be much better, # 2

a service rivals institutions?

Companies can be rivals, of course. (Although the word is not used in a positive way in business AFAIK). Services can be unrivalled but it's not the right word in this context IMO

no - but the rather clumsily worded OP should read "Let us see how this service rivals the service provided by traditional institutions in ordinary matters." for example. The services provided are being compared - as I see it.  "service provided" is implied (#3).  

agreed #13, your suggestion is fine of course.

Just wanted to make it plain and clear that the original sentence needs to be amended.