Math: Perception of only or mostly doing research in Master's for Ph.D. admission folks
Background: I am the director of graduate admissions in the pure math department of my university.

In my experience, we basically ignore "research" conducted by undergraduate and masters students (unless is it truly remarkable, in which case you will almost certainly not need advice on the internet to get into a strong PhD program).  It just isn't a good predictor of whether or not you will be successful in a PhD program.  You will get very different advice from people who are not mathematicians, but you should ignore it -- graduate programs in math are very different from graduate programs in the other sciences.  This advice also might not apply if you are applying to departments of applied mathematics.

What we care about are 


The courses you have taken, and
Your grades, and
Your letters of recommendation, and
Your GRE scores.


The above list is un-ordered; different members of the committee weight the above factors differently.