If your institution has a license, I think IPA is the best option (but it isn't free).

IPA has a mix of known and predicted targets. You can download the predicted targets for free (http://www.targetscan.org/). Many of the known interaction may come from IPA's proprietary database, but I think you can also find known interactions from tarbase (http://diana.cslab.ece.ntua.gr/tarbase/).

Once you have text files with the miRNA-mRNA interactions, you can search for which interactions fit your data (based upon genes with with opposite fold-change / log2ratios and/or a negative correlation between miRNA and mRNA expression levels).

You also can use starBase ( http://starbase.sysu.edu.cn ). starBase Platform is designed for deciphering Pan-Cancer Networks of miRNA-mRNA, miRNA-lncRNA, miRNA-ncRNA and protein-RNA by mining clinical and expression profiles of 14 cancer types (>6000 samples) from The Cancer Genome Atlas (TCGA) Data Portal. It explore the negative correlation between miRNA and mRNA expression levels using expression profiles ( http://starbase.sysu.edu.cn/targetSite.php ).

Maybe you will find the newer release of MAGIA helpful: http://gencomp.bio.unipd.it/magia2

You can try InCroMAP software. It is based on Java, but user - friendly, and free, of course. Basic functions:

• analyzing data: mRNA expression data, miRNA expression data, DNA methylation data, and protein modification expression data
• you can analyze a single dataset: i.e. enrichment analysis, pathway - based data visualization, visualization of miRNA targets, metabolomic overview
• and the main function: integrated cross-platform microarray data analysis - something like MAGIA does, but more expanded and updated.

You can try and read about it here and here.

Best regards!

Magda