What are some applications of Steenrod operations (or similar constructions) in algebraic geometry?
I am dimly aware of the the use of these Voevodsky's work on motivic cohomology, and would be happy if someone explained that in a comprehensible way. However, the primary motivation for asking the question is to see if such operations on the coherent (or de Rham) cohomology on algebraic varieties (in positive/mixed characteristic) have concrete geometric consequences.
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.
You may be interested in the thesis of Olivier Haution, entitled Steenrod operations and quadratic forms. The author gives a new approach to constructing Steenrod operations on the Chow ring mod p, and uses this to give prove a theorem on the parity of the Witt index of quadratic forms.
There is also the work of Sasha Vishik on symmetric operations in algebraic cobordism (available from the author's web page) which has applications to the rationality of cycles.
