Let's pretend that we know local Langlands at a fairly high level of generality... i.e. we know something along the lines of:

Let $G=\mathbf{G}(F)$ be the group of $F$-points of a connected reductive algebraic group $\mathbf{G}$ defined over a nonarchimedean local field $F$ with separable algebraic closure $\bar{F}$. Let $W_F'=W_F\times SL_2(\Bbb{C})$ be the Weil-Deligne group of $\bar{F}/F$ and let $^LG={}^LG^0\rtimes\mathrm{Gal}(\bar{F}/F)$ be the Langlands dual of $G$, where $^LG^0$ is the connected reductive complex algebraic group with root datum dual to that of $G$.

Then there exists a natural surjective map

$$\mathrm{Irr}(G)\twoheadrightarrow\mathrm{Hom}(W_F',{}^LG),$$

where $\mathrm{Irr}(G)$ is the set of equivalence classes of smooth irreducible complex representations of $G$. This map has finite fibres (the $L$-packets), is "compatible" with a list of operations: parabolic induction, twisting, etc, and is the unique such map.

As far as I know, this is a theorem (or maybe very close to being one for the latter two?) for $GL_N$, $SL_N$, $Sp_{2N}$ and $SO_N$.

When I'm trying to explain why I'm interested in this to someone I'll give the usual explanation along the lines of "we want to understand $\mathrm{Gal}(\bar{F}/F)$, local class field theory lets us understand the abelianisation of it in the form $F^\times\simeq W_F^{\mathrm{ab}}$, LLC generalises the dual form of this to a nonabelian setting and should tell us an awful lot about $\mathrm{Gal}(\bar{F}/F)$". Obviously you aren't going to hit them with the correspondence as stated above, but you can usually get away with saying "smooth irreps of $GL_N(F)$ naturally correspond to $N$-dim complex reps of $W_F'$, and that should generalise in a reasonable way to other groups".

At this point, I'll usually have either satisfied my questioner, or they'll ask me if I can give an example of what the LLC should let us do. That's when I run in to trouble -- I don't know of a single, reasonably simple, appealing application of it. In the global case people often bring up the proof of FLT. This isn't exactly "simple", but it's at least well known and can be summarised as "if FLT doesn't hold we have a non-modular semistable elliptic curve. Wiles then uses Langlands-Tunnell as a starting point, does a lot of work and eventually shows that every semistable elliptic curve is modular, hence FLT".

So... are there any such good examples of applications of the local correspondence?

representation-theoreticcriteria of Neron-Ogg-Shafarevich and Grothendieck). The last chapter of Katz-Mazur gives ageometricproof of surprising reduction properties for certain quotients of Jacobians with bad reduction. That proof (using vanishing cycles based on delicate bad reduction of modular curves) feels like a miracle. By contrast, LLC and local-global compatibility for GL$_2$ lead to an entirely different proof which, while not easy, doesn't have the feeling of a miracle. $\endgroup$ – user27920 Jun 15 '14 at 10:29