I used to think that the entire theory was intellectual masturbation, but two examples in particular completely changed my mind.
The first is the Pontryagin-Thom construction, which exhibits an isomorphism between the $k$th stable homotopy group $\pi_{n+k}(S^n)$ and the framed cobordism group of smooth $k$-manifolds. This is even interesting (though more elementary) in the case $k = 0$, where it recovers the basic degree theory that you learn in your first course on topology. This was originally developed by Pontryagin to compute homotopy groups of spheres, but now it is regarded as a tool in manifold theory. These matters are discussed in Chapter 3 of Luck's book on Surgery theory, for example.
The second application is to physics. Unfortunately I don't understand this story very well at all, so I'll begin with what I more or less DO understand (which may or may not be well-known). The basic idea begins with the problem of situating electromagnetism in a quantum mechanical framework. Dirac began this process by imagining a "magnetic monopole", i.e. a particle that would play the role for magnetic fields that the electron plays for electric fields. The physical laws for a charged particle sitting in the field determined by a magnetic monopole turn out to depend on a choice of vector potential for the field (the choice is necessarily local), and Dirac found that changing the vector potential corresponds to multiplying the wave function $\psi$ for the particle by a complex number of modulus 1 (i.e. an element of U(1)). If we think of the magnetic monopole as sitting at the origin, then these phases can naturally be regarded as elements of a principal $U(1)$-bundle over $M = \mathbb{R}^3 - \{0\}$. But $M$ is homotopy equivalent to $S^2$, and principal $U(1)$-bundles over $S^2$ are classified by $\pi_1(U(1)) = \mathbb{Z}$. Proof: think about the Hopf fibration. The appearance of the integers here corresponds exactly to the observation of Dirac (the Dirac quantization condition) that the existence of a magnetic monopole implies the quantization of electric charge. It is remarkable to note that Hopf's paper on the Hopf fibration and Dirac's paper on magnetic monopoles were published in the same year, though neither had any clue that the two ideas were related!
The story goes on. The so-called "Yang-Mills Instantons" correspond in a similar way to principal $SU(2)$ bundles over $S^4$, which are classified by $\pi_3(SU(2)) = \mathbb{Z}$. Again, the integers have important physical significance. So these two classical examples motivate the computation of $\pi_1(S^1)$ and $\pi_3(S^3)$, but as is always the case this is just the tip of an iceberg. I am not familiar with anything deeper than the tip, but I have it on good authority that physicists have become interested in homotopy groups of other spheres as well, presumably to classify other principal bundles (it seems like a bit of a coincidence that the groups which came up in these examples are spheres, but maybe one reduces homotopy theory for other spaces to homotopy theory for spheres). People who know more about physics and/or the classification of principal bundles should feel free to chime in.
A great reference for the mathematician who wants to learn something about the physics that I discussed here is the book "Topology, Geometry, and Gauge Fields: Foundations" by Naber.
SERGEI O. IVANOV, ROMAN MIKHAILOV, AND JIE WU have recently(2nd June 2015) published a paper in arxive giving a proof that for $n\geq2$, $\pi_n(S^2)$ is non-zero.
You can look at it in the following link.
Sergei O. Ivanov, Roman Mikhailov, Jie Wu, On nontriviality of homotopy groups of spheres, arXiv:1506.00952
Best Answer
My apologies for updating this very old question.
As already mentioned, the 31st homotopy group of $S^2$ is the same as the 31st homotopy group of $S^3$. Serre's mod-C theory shows that this is a finite abelian group, and moreover that for any prime $p$ the first $p$-primary torsion occurs in $\pi_{2p} S^3$. This means that we don't have to check any primes $p$ for which $2p > 31$, which leaves us with the primes $p=3,5,7,11,13$.
The main tool for calculating these is the EHP spectral sequence. For $S^3$ the EHP sequence is pretty simple: there is a long exact sequence $$ \dots \to \pi_{n-1}(S^{2p-1}) \to \pi_{n}(S^3) \to \pi_{n}(S^{2p+1}) \to \pi_{n-2}(S^{2p-1}) \to \dots $$ that goes through $n > 3$.
The next point is the stable range. At odd primes and for odd spheres, the stable range is larger than that given by the Freudenthal theorem: if $n$ is odd, the stabilization map $\pi_{n+k}(S^n) \to \pi_k^S(S^0)$ is an epimorphism if $k=(n+1)(p-1) - 2$ and an isomorphism if $k < (n+1)(p-1)-2$. This means that all the groups relevant to computing $\pi_{31}(S^3)$ appear in the stable ranges of $\pi_*(S^{2p \pm 1})$ for $p \geq 5$, and those stable ranges are moreover very sparse (they only include the "image of J", which is well-known). If I have calculated correctly, $\pi_{31}(S^3)$ includes no $p$-torsion for $p \geq 5$.
That leaves only $p=3$, where we have to switch to a different algorithm ("I cannot do this, so I better look up what Toda calculated"). Toda showed that the 3-primary part of $\pi_{31}(S^3)$ is $\Bbb Z/3$. I found this in his 2003 text, "Unstable 3-primary Homotopy Groups of Spheres", but that was simply because I had it handier.
Combining this with the information from Mark Grant's answer gives $\Bbb Z/3 \oplus \Bbb Z/2 \oplus \Bbb Z/2 \oplus \Bbb Z/2 \oplus \Bbb Z/2$.