Since a long time I have been thinking of creating an encyclopedia of ignorance for the courses that I teach. When I correct papers, I see some standard mistakes students make.
Let me start randomly collecting them. May be some day I shall put them in order. I think I shall keep updating this page.
Physics:

 The and field in an electromagnetic wave are point quantities. Text book diagram often makes people think about them as quantities extended in space.
 Density is also a point quantity which is defined as the ratio of mass ina volume to the volume, in the limit volume going to zero. Though theoretical such definitions are very useful practically.
 It is possible to have the function value zero at a point while its gradient is nonzero.Which also implies that the magnetic field at a point may be zero but the force on a dipole at that point may be nonzero.
 Keeping a body magnetically levitated does not require work.
Mathematics:
 is not the same as , many student wrote this in their answers in the last exam. I tell them this would be like saying that usually makes sense to them.
 Over and over again I have to remind the students to try to make sense of the equations they write. For example while applying Lagrange’s mean value theorem to on students wrote, , replacing on the RHS by . The formula when simplified gives, which should raise a serious doubt, provided they stop doing mathematics mechanically and start thinking about meaning of the formulas.
 Exponential processes in a variable x can also be of the form , somehow many students think that only would be an exponential process but any given number can be written as hence can be expressed as .