Essentially antibodies are developed in life animals as a consequence most antibodies are directed against denatured proteins (which also have a 3D shape). The problem of antibody binding after SDS PAGE is not the denatured state but the SDS bound to the target. You have to get rid of that.
Monoclonal antibodies are often less useful in immunoblotting since they only react with a single epitope. In my experience using polyclonal antisera is much better. You then have a chance that some of the antibodies will bind.
It depends on antibody types. I have been characterizing both monoclonal and polyclonal antibodies from human, mouse and hamster. Linear/continuous/sequential and conformational/native/3D epitopes are critical for antibody binding. WB is unable to detect the 3D one, even native gel may fail if the directed epitope has some protein modification and same goes to ELISA, however, it depends on fixation methods for ELISA. To detect 3D binding, best would be IP and needs confirmation with a linear antibody. Unfixed fresh frozen tissue section (5micron) is a good for conformational one. Another choice is MALDI-TOF/MS-MS for the 3D type. For non-specific bands on SDS-PAGE gel with different species, you need to do absorption assays to see the reduced level of binding. For receptor or extracellular ligand molecules, you can run FACS. So one assay cannot give you the final or confirmatory answer for the nonspecific reactivities. Also keep in mind that concentration of both antigen and antibody is an important regulator for proper binding as they may impose prozone phenomenon. Thanks.