Why don't magnetic monopoles exist? Electric monopoles do exist, but magnetic monopoles don't. Why?
The question is closed because I need to clarify it, but I don't know how I could ask it another way. However, I've recieved many answers that were appropiate and added something to my knowledge, so I consider it answered.
 A: Within the framework of standard model (SM) magnetic monopoles are non-existent. It is quite subtle as to why this is not the case. To begin, look at the Dirac's famous charge quantization condition. It was first pointed out by Dirac that on the quantum level the existence of the monopole will lead to the 
\begin{equation}
qg = 2\pi n \qquad\qquad \text{where n is an integer.}
\end{equation}
Where $q$ ang $g$ are the electric and magnetic charge (monopole) respectively. Now just consider the electric charge quantization within the framework of SM, which is not possible to achieve. Thats because in SM electric charges are the eigenvalues of $U(1)_{\text{em}}$ generator Q. Point is eigenvalues of $U(1)_{\text{em}}$ generator are continuous (moving on a circle) where electric charge of leptons and quraks are quantized. This simply implies that one can't get any satisfactory explanation of electric charge quantization from SM. Consider this to be true, existence of monopole (following Dirac's quantization condition) is SM also not possible. 

Topological considerations lead to the general result 
that stable monopole solutions occur for any gauge theories in which a simple 
gauge group G is broken down to a smaller group $H = h \times U(1)$ containing 
an explicit $H = h \times U(1)$ factor. For a review of the topological arguments see (Coleman 1975, 1981). Clearly this is compatible with the fact that expectation of charge quantization and existence of monopole are related and that charge  quantization follows from the spontaneous symmetry breaking of a simple gauge group. In the grand unified theories where the symmetry is 
broken from some large simple group, e.g. $SU(5)$, to $SU(3)_{\text{c}} \times SU(I){\text{em}}$, there are also monopole solutions of the 't Hooft-Polyakov type. The monopole mass is determined by the mass scale for the symmetry breaking $M_{X}$ (mass of the color changing gauge bosons). In the $SU(5)$ GUT, $M_{X}\equiv 10^{15}$GeV. This means that this type of monopole is out of reach for its production by accelerators (LHC energy scale is 14 TeV). So even if any monopole do exist, its out of our reach. Keep in the mind that $SU(5)$ has already ruled out!
A: There is no theoretical reason why magnetic monopoles cannot exist and indeed there are good reasons for supposing that they should exist. It's just that we have never observed one. In the past there have been various experiments to detect magnetic monopoles, though I think everyone has given up on the idea by now.
If you're asking why we can't get monopoles out of a magnet that's because the magnetic field of a magnet is built up from the individual magnetic fields of the unpaired electrons in the magnet, and those electrons have a dipole field. There isn't any way to combine the dipole fields of the electrons to create a monopole, though it's possible to make things that look locally approximately like monopoles.
A: I think that John Rennie's is a good answer, specially due to the link given where it is shown the reasons why physicists expect the magnetic monopoles to exist. However I would like to stress that this is not simply the case of "explaining the non existence of something that we never saw". It is rather how to explain the lack of observation of something that it is expected to exist.
The strongest reason they are expected to exist is the fact they are predicted in Grand Unified Theories (GUT's), such as $su(5)\rightarrow su(3)\oplus su(2)\oplus u(1)$ or $so(10)\rightarrow so(6)\oplus so(4)\rightarrow su(3)\oplus su(2)\oplus u(1)$. They naturally appear as localized and stable solution to the equations of motions for the theory. 
So a natural question to do is: Why do not we observe these theoretically predicted monopoles? Once they are predicted by a consistent theory we no longer can answer "they don't exist because they are not observed". We do need to explain why we do not see them even though the theory says they exist.
A simple but not interesting answer would be that the Universe did not show a spontaneous symmetry breaking patterns which allow for stable monopoles (such as the above ones). 
A much more interesting scenario is the following one, called monopole problem. The monopoles predicted by GUT's are in general heavy and abundant. Its abundance is calculated to be far greater than the observed density of the Universe. This was one of the Alan Guth's motivation for proposing an Inflationary Universe. The idea is that these far too many monopoles were diluted during an infationary epoch.
A third explanation which have been proposed is that there was an intermediate phase during which monopoles anihilate themselves by means of flux tubes, which are other topological solutions predicted by GUT's.
A: The historical basis of this belief is embodied in Gauss' Law:
$$ \nabla \cdot \mathbf{B} = 0 $$
This form is widely accepted for classical electromagnetism (as opposed to the form modified to allow for magnetic monopoles). It implies that the net magnetic flux over any surface is zero. A magnetic monopole would cause the magnetic flux of a surface to be non-zero, and so it would violate this law.
It correctly predicts the results of experiments where classical physics can be applied. As with all physics, this formula is subject to new theories and experiments, but it is correct enough to become widely accepted as part of classical physics. This is one of Maxwell's equations, though he used a different form.
