As a Master Thesis supervisor, how should I assist students with poor fundamental skills (e.g., problem solving)?

Voevodsky's proof of the Milnor (and Bloch-Kato) conjectures uses, in a vital way, the Steenrod operations in motivic cohomology. I apologize first for being vague but I will give a reference at the end!

Roughly speaking it's used in the following way: so the Milnor conjecture states an equivalence $A(k) = B(k)$ which depends on fields $k$. What one first proves that if one extends $k$ to be ``big enough" (no extension prime to the characteristic we are interested in and some conditions on the Milnor $K$-theory) then the statement is true by some (Galois cohomology) computations.

So now if one has a counterexample to the Milnor conjecture then one wants to show that the counterexample "propagates" - in other words it is still a counterexample after one extends the field (this has to be done in a clever way and this is where the algebraic geometry comes in through the "Rost quadrics"). One then extends the field to the range where we already know that the statement holds and get a contradiction.

The point of the Steenrod operations is to show that the counterexample "injects" into this extension of fields - what happens is one uses the motivic analogue of the Margolis element to map these counterexamples into each other and Voevodsky used pure topological methods to show that these Margolis elements act injectively. The sketch of this idea is beautifully written in Dan Dugger's notes and they are available here: http://arxiv.org/pdf/math/0408436.pdf.

Not sure if this counts, but a much older application of $Sq^1$ (appearing as a Bockstein) and some of its cousins appears in Serre's paper on Witt vector cohomology. He constructed a family of operations, in characteristic $p$, $\beta_r: H^1(\mathcal{O}_X) \rightarrow H^2(\mathcal{O}_X)$ called the higher Bocksteins and it's reasonable to think of the first of these as a $Sq^1$.

A concrete application was given by Mumford who showed that a surface over a field of positive characteristic has smooth Picard scheme if and only if all of the Bocksteins on $H^1(\mathcal{O}_X)$ vanish. He later used this fact (amidst many others) in his and Bombieri's classification of algebraic surfaces in positive characteristic.

+ and - work here.