Applied/Pure Math PhD Coursework
It might be somewhat controversial for some here, but I don't think there is any one course that you should take in order to get in a particular graduate program. The reason behind my statement is that I believe the logic presented in the OP is rather backwards; one usually pursues graduate studies in a particular field that s/he is knowledgeable and interested in. In other words, you get your undergrad and Masters, and based on what you know and like, you apply to PhD programs in fields where you are competent. The other way around (deciding on a PhD program without have the undergrad and masters done, and choosing courses based on the desired PhD program) does not make much sense, in my humble opinion. 

Also consider that "a PhD in pure/applied maths" is really ill-defined. I can think of a hundred different projects that would have different requirements, with regards to previous courses. I would recommend deciding on a specific subject that you find interesting e.g. "elliptic curve cryptography" or "convex optimization" etc (not just pure/applied math).. then look for announced PhD programs based on projects focused on these subjects of interest. 

All that being said; I would think  advanced level courses in matrix theory (decompositions etc), functional and complex analysis, as well as  optimization theory would be useful in many different graduate programs at most maths departments.