Practically speaking, in the world of software it doesn't really matter which you use.
When speaking of empty result sets we use the plural:
the lab came up with zero results.
We also pluralize result sets other than one:
the labs came up with differing results.
However, we don't pluralize a single result:
the lab came up with one result.
Your discomfort seems to stem from the fact that your statement contains both a plural and a singular integer: 0 of 1. It might be useful here to look at how we would comfortably speak of other result sets.
1 of 5 results were shown to be false.
Here it causes us no consternation to speak of five results even though there is a singular integer involved. This is because the grammatical number is predicated on the quantity assigned to the object: 5 results.
In fact, we are used to hearing such assignments all the time.
4 out of 5 dentists recommend Crest toothpaste.
Two out of three workers are not saving enough for retirement.
So it is customary for the ear to hear the object of those statements pluralized. And what is customary can be hard to shake. So when we see
Displaying 0 of 1 result
in the output of a search, it makes us a bit uncomfortable. Since there is only one result set that contains the number 1, we rarely hear it.
And even though I believe that "0 of 1 result" is the grammatical way to go, I'm going to advise you at this point that it really doesn't matter. This falls under the banner of what I would call "computerese." Computers don't use English the way people do, at least not all the time. And we allow for the exceptions, for the most part, without really thinking about them.
When I started programming for a living I was a bit obsessed with outputting grammatical responses in my programs. I went to some degree of trouble to make sure that my programs output "You have 1 response" where appropriate instead of the "You have 1 responses" I would see in other people's software. Then I had to make those fine adjustments everywhere, and to maintain that code, and have other people maintain that code, and sometimes those others would decide such fine distinctions were too much trouble and would take them out. Then there was the problem of what to do about irregular plurals in some nouns. A pluralization class would be near impossible to write. It became more than a battle: a never-ending war for the sake of a concept of grammatical number that I alone seemed to be waging.
Eventually I called a truce. This was just common sense, given the fact that very few people were troubled when a software program seemed to mess up grammatical number in a small edge case. They just accepted the results, understood them without trouble, and went on with their lives. So did I.
Conclusion: Computers don't talk the way we do, and we have come to accept that.