I cannot speak from the point of view of a Math major in US since I never was one. I completed my undergraduate studies in engineering and currently pursuing a Ph.D. in pure mathematics. In my opinion, applied mathematics (though admittedly this quite a generic term) would be more accessible to an undergraduate considering research than pure Mathematics. I ended up publishing two single author papers in respected journals while in my senior year. I had started working on both these problems during my junior and both of them were picked by me. When I though I had a good insight into the problems, I approached the faculty within my university for suggestions. I think it is safe to say that a lot of problems in applied mathematics require less sophisticated machinery than is used by most pure mathematicians. Many of my engineering friends started working on their Ph.D. thesis problems fresh out of a Bachelors in areas which could be termed as applied mathematics. This contrasts with most pure math grad students I know who usually spend between 1 to 3 years of coursework before starting to work on a concrete research problem. So it seems that "undergraduate level coursework" would be sufficient in handling a good number of applied math problems. So if you are advanced undergraduate student with a good background in one such allied area, I think it might be worthwhile to explore this possibility. After all you can gain valuable experience doing research even if you do decide to pursue some other area of math in your graduate life.