Monopoles are still created in inflationary models. They're just created before (or during) inflation, so that the rapid expansion thereafter dilutes their density to unobservably low levels.
At the time when the monopoles are created, they're created at a density of order 1 per Hubble volume -- that is, there's one in each "observable Universe" at that time. In general, when a symmetry breaks, topological defects form that are separated on a length scale of order (speed of propagation of the field)(time scale over which the symmetry breaks). The first is of order $c$, and the second is of order the Hubble time, so monopoles are separated by a distance of order the Hubble length.
You should take "of order" here very liberally -- I don't actually care if I'm off by factors of $10^5$ or $10^{10}$ or anything measly like that! After all, inflation blows up lengths by something like $10^{20}$ or more. So one monopole per horizon volume becomes one per $10^{60}$ horizon volumes. (Also, the horizon volume continues to change after inflation is over, but not by anything like this sort of factor.)
With densities like that, we certainly wouldn't expect to see any monopoles. Problem solved.
Many years ago I considered the situation of a genuine monopole continually threading through the middle of a wholly superconducting loop. So we have two interlocking Roman rings - one an electric charge circuit, the other a magnetic charge circuit. Depending on the relative sense of circulation, either the monopole gains energy at the expense of the supercurrent, or vice versa. Well actually, it might not be that simple.
Thing is, superconductivity is intimately associated with the usual vector potential A, and a supercurrent will only change in response to a change in an externally applied A. Such as to maintain the line integral of net A around the supercurrent invariant. But A is only generated by moving electric charge. The hypothetical 'back emf' of circulating monopole would be owing to an E field the analog of the B field of moving electric charge. On a time-average basis it would be steady given a steady monopole current. Hence of a fundamentally different character to an $E = -dA/dt$ owing to time-varying electric current, that the supercurrent would know and respect. Hence regardless of whether circulating monopole gains or loses energy in following along the lines of B generated by the supercurrent, the supercurrent itself will do squat. There is a similar dilemma when it comes to the predicted net force/torque balance - or rather imbalance.
Upshot is, one either accepts that energy-momentum conservation would dramatically fail, or take the scenario as proof that a genuine monopole cannot exist!
Best Answer
Luckily, there exists a truly amazing one-to-one correspondence between magnetism and electricity.
Monopole in magnetism is analogous to charge in electrostatics/electricity. Just like we have two types of charges (+ve & -ve), we have two types of monopoles (north & south). The north magnetic pole is also known as the positive pole and the south magnetic pole is known as the negative pole.
We refer to the size/magnitude of the charge as charge itself, but for a monopole, we refer to its size/magnitude as pole strength.
Electric charge is denoted by the letter 'q', the pole strength of a magnetic pole is denoted by the letter 'm'.
Electric charges produce electric fields. The magnetostatics analog is the magnetic field.
We make use of electric field lines to represent the electric field visually. By convention, electric field lines start from a positive charge and terminate at a negative charge. So is the case with magnetic poles as well. Magnetic field lines originate from the north pole (+ve pole) and terminate at the south pole (-ve pole).
The strength of the magnetic field produced by a monopole is given by
$$\vec{B} = (\frac{\mu_o}{4\pi})\frac{m}{r^3}\vec{r}$$
You probably noticed that the equation is identical to Coulomb's law except for the fact that we have magnetic pole strengths instead of the magnitude of charge.
As of 2016, we aren't sure if magnetic monopoles exist (we haven't found one yet). When we try to make a monopole by slicing a bar magnet exactly in the middle, we end up creating two new bar magnets where each magnet has both north and south pole.
Well, you might ask how do I know about the behavior of monopoles if monopoles don't exist. Whatever I said above is just a hypothesis and the hypothesis is consistent with the reality.
When two magnetic poles of equal pole strength are kept close to each other, we call it a dipole. A bar magnet is an example of a dipole.
We have an electric dipole moment equivalent in magnetostatics. We call it magnetic dipole moment (usually represented by '$\vec{M}$').
$$M = md$$
where d is the distance between the two poles and m is the pole strength of the poles. The direction of the magnetic dipole momentum is from negative pole to the positive pole (In the case of an electric dipole, it is from the negative charge to the positive charge).
The field along the axis of the magnetic dipole is given by,
$$B = 2(\frac{\mu_o}{4\pi})\frac{M}{r^3}$$
and the field along the equatorial line of the dipole is given by
$$B = (\frac{\mu_o}{4\pi})\frac{M}{r^3}$$
If you open your textbook and look for the formulae of the electrostatics analogue of the dipole, you will find that the formulae are a perfect match.
The formulae given above can also be derived from the magnetic monopole hypothesis.
The similarities don't end here.
Torque on an electric dipole in an external uniform electric field is given by
$$\tau = \vec{p}\times\vec{E}$$
Torque on a magnetic dipole in an external uniform magnetic field is given by
$$\tau = \vec{M}\times\vec{B}$$
You shouldn't be surprised if you find more exact matches. The magnetic monopole behaves just like electric charge and the formulae are identical. The derivations for magnetic dipoles are identical to derivations for the electric dipoles so you should end up with the same formulae.
A circulating electric current behaves like a magnetic dipole.
If you derive the formula for magnitude of magnetic field along the axis of the circular coil, you should get something similar to the following,
$$B = \frac{\mu_o NiR^2}{2(R^2 + x^2)^{\frac{3}{2}}}$$
where $N$ is the number of turns in the coil, $i$ is the current passing through the wire, $R$ is the radius of the coil and $x$ is the distance along the axis of the coil.
If we use the approximation $x>>R$, we get,
$$B = \frac{\mu_oNiR^2}{2x^3}$$
Rearranging further, we get
$$B = 2(\frac{\mu_o}{4\pi})\frac{(iN\pi R^2)}{x^3}$$
If you look carefully, you will notice that the formula looks very similar to the formula which gives the strength of the field along the axis of a magnetic dipole.
We can define magnetic dipole moment as
$$M = iN(\pi R^2)$$
The direction of the magnetic moment is the direction of the magnetic field at the center of the coil.
Now the formula looks pretty much identical to the formula given earlier for the strength of the field along the axis of a magnetic dipole.
In general, magnetic moment for any closed loop circuit can be defined as
$$\vec{M} = iN\vec{A}$$
where $\vec{A}$ is the unit normal vector for the loop.
One of the consequences of monopoles not existing is that magnetic field lines are always closed. Consider any dipole, a field line which emanates from the north pole must end up back at the south pole.
The number of field lines that leave the surface is equal to the number of field lines entering the surface. Hence, there is no net flux entering or leaving the surface.
$$\oint{\vec{B}.d\vec{A}} = 0$$
In the Gauss Law for electrostatics, we consider the volume charge density or the net charge enclosed within the gaussian surface. The same goes for Gauss Law for magnetism.
$$\oint{\vec{B}.d\vec{A}} = \mu_o m_{enclosed}$$
since there can never be any isolated monopole, $m_{enclosed}$ is always zero.
If monopoles existed, then $m_{enclosed}$ needn't be zero. Hence, the Gauss law would be invalid.
If we ever find a monopole, it would imply that Gauss law for magnetism would be invalid. We will need to amend the law to make allowance for monopoles. (the amended version of the law was presented in the answer to your previous question)
Discovery of magnetic monopoles would imply the existence of magnetic currents. We would have a new chapter in our physics textbooks. Possibly a new engineering branch might appear (magnetronics?).
Every electron has two types of magnetic dipole moment, namely spin magnetic dipole moment and orbital magnetic dipole moment.
There are several electrons in an atom and most electrons cancel out each other's magnetic moment. When all the electrons cancel out neatly, the total magnetic moment of the atom is zero. The substances where the total magnetic moment of the constituent atoms is zero are known as diamagnetic substances.
There are many substances with unpaired electrons. This gives rise to a net magnetic moment for each atom. When these atoms collectively align in the same direction. This kind of collection of atoms is known as a domain. When domains align to give a net magnetic moment to a substance, the substance is said to be ferromagnetic.
Discovery of a magnetic monopole shouldn't affect the existing theories. However, it might force us to investigate further at the fundamental level.