How to predict the top microRNAs that bind to 3`UTR of a specific gene
You'll find TargetScan quite useful. You can pick from a number of species and simply enter a gene name. It will then get the 3' UTR, show its conservation across species (binding sites in a conserved area are more likely to be active), find miRNAs that can bind to the 3'UTR, and then split those up by whether they bind to highly or poorly conserved regions. It also categorizes the hits according to various properties, like proximity to the open reading frame, in case you want to perform any sort of filtering by that.

Edit: BTW, you may want to take the target miRNAs and look at their expression in your tissue of interest. That's a good way to further narrow down their relevance.