Is it OK to be picky about math you find interesting? I am a layman interested in mathematics, and I would like to hear mathematicians' views on the following: Is it normal to be picky about mathematical stuff you find interesting?
I ask because 80% of math I encounter does not seem interesting to me. I want to know if this is true for most people or if there is something wrong with me.
 A: At any particular moment, or for stretches of time, it is entirely reasonable to find various things boring... or incomprehensible, or both.
But/and as one learns more, one's opinions will inevitably change.
And, yes, admittedly, many published papers are contrived, with the main goal of having a publication to make a good impression on dept heads, deans, and funding agencies. No real intrinsic interest... and it's hard to tell, from an amateur's or novice's viewpoint.
So I'd recommend that you (and everyone...) follow their own tastes and curiosity, BUT be open to the possibility that things that seem boring or pointless or incomprehensible at one moment may later seem quite otherwise. Don't pretend to make permanent decisions...
A: I'm an undergraduate in mathematics. There are certain modules that I look forward to and others that I dread. Liking the entirety of mathematics is pretty unlikely.
You are normal, congrats!
A: You wade into the realm of personal taste.  I'd say math is like wine:  different things appeal to different sorts.  
A: I would say that this highly depends on the level of your education and your goals.  If you are a postdoc or an advanced graduate student, with a very strong math education under your belt, you are somewhat free to say that certain areas are not as interesting to you as others.  You have some idea of what the kind of field you want to pursue, and which techniques may or may not be useful to you.  There is a huge amount of math out there, much of it very specialized and probably not interesting to mathematicians outside of that particular field.  Of course, even professionals should be cautious about dismissing fields of mathematics - mathematicians are very often surprised to see that techniques they may have otherwise dismissed turn out to be useful.
However, if you are pursuing a general math education but you are not currently an expert, you should be very careful about dismissing math as uninteresting.  It would be a big mistake to say "linear algebra is boring, so I won't bother."  Linear algebra is useful for virtually every subfield of math - it's a very important part of any mathematicians' education.  These basic fields should be studied even if they seem uninteresting at first glance, because they are indispensable.
If you are not pursuing a general math education - e.g., you are studying it as a hobby, or you have a very specific goal in mind that you know requires only a specific subset of the typical coursework - then you are more free to pick and choose to study a few things that are interesting to you.
A: If you are contemplating a math degree, or any other degree for that matter, I would advise a much higher interest rate than 20%. Life is too short to spend time on something that uninteresting. While a 100% interest in all math topics is unlikely, I would say that the mathematicians I know, and probably most of the people answering questions here, have a very much higher interest rate.
A: I would say it would be unwise and rather odd for a mathematician to dismiss any part of the subject matter of mathematics as uninteresting. The subject is very large and there are deep connections between apparently very different aspects: as a simple example, consider the many different proofs of the fundamental theorem of algebra. If 80% of the maths that you encounter seems to be uninteresting to you and you can't see why that 80% of interest to anyone else, then I don't think maths is for you. (I'm personally not very excited by the the theory of differential equations, but I know it's an important subject and I appreciate all the work that's been done on it over the centuries.)
