if it's non significant is like zero... no correlation... whatever the result, if no significant it was obtained by chance. In one of  my studies with microsats, I did not have a significant correlation in a very small geographical scale (within village), although for the species there is correlation on a larger scale. Possible explanations were related to factors other than geographic  distance having more impact in the differentiation of the populations. Which makes total sense if you are working in a scale that falls within the dispersion range of the species.... if the organism could move within an area of 1000mts, the analysis of populations within that distance is like within a panmitic unit (although with some stratification) . Analyzing other factors involved in the history of the population I was able to come up with a very reasonable hypothesis to explain the pattern. 

In my opinion, you can consider it non-significant. In principle I could suggest also to check if you have all samples from same species, otherwise you certainly can have negative correlation if you took two neighboring populations of different species and add third from remote locality with belongs to one of them. Might be there are additional possible explanation, difficult to say without seeing data.

Dear Arvind: That is the purpose of significance tests. Significance depends not only on the correlation values, but on the number of samples etc. Anyhow, if the test is non-significant, you can not assume there is correlation.

Hope this helps

Theoretically speaking it may happen  and even be significant if you have 2 or more natural groups occurring in some N localities, and you group the samples for calulation of Fst. If your grouping is incorrect, that is it does not fully coincide with the natural one, and if your localities are ordered geographically, you will have it.

At risk of overlapping with the previous answers, I'll chiem in here. The way I see it, there are two things to respond to in your post. 

1. Yes, it is possible that Fst could have a negative spatial relationship. That is rather atypical, but plausible. Genetic differentiation among populations does not always decrease with distance (though typically it does). 

2. You should interpret the pattern in your dataset as lacking any such correlation, given the results of your analysis. 

Hope that helps. 

I can think of a theoretical example where over a given spatial scale this would be possible.  If juveniles of a species have a minimal (and large) dispersal distance behaviourally programmed, and the landscape is significantly fragmented, then it would be possible with a given level of fragmentation and sampling scheme to get a significant negative correlation between FST and spatial distance (and this would have biological meaning).  This all depends on very precise and linked values of dispersal distance, fragmentation and sampling distances, so I would think this is highly unlikely.  

As others have already said, non-significant = no correlation.  If a statistical test is not significant you should not report it as anything other than non-significant.  It doesn't matter how interesting the pattern, non-significant indicates you have insufficient evidence to support any correlation seen. 

Thanks all for your valuable input....The area of my study has very complex topography with vast number of small and large mountain ranges, thick forest cover and rivers, I think all these might be contributing in negative values.

Definitely you can not consider the correlation, since it is not significant. Anyway, thinking about the reasons behind a result is important, even if the result is not supported by statistics. I think that the assumption of linear correlation between geographic and genetic distance is not always true, many factors might play an important role in giving different patterns of genetic differentiation among populations. Sometimes very near populations are more differentiated from each other than with very far ones. It depends on the dispersion ability of the individuals, on the presence of ecotones or other dispersion barriers, etc. Genetic is interesting also for these reasons...Unfortunately, no statistical significance= no scientific value, therefore you can not use this result for publication purpose, but nobody can block your thinking. You might find other exciting results starting from this non-significant result.

Good luck with your study!