Actual shape of electromagnetic waves The electromagnetic waves are popularly represented as two sinusoidal waves at right angle to the direction of wave motion. But it is only a representation and not an actual shape right? Let me explain to clarify my question...
Let us say the blue sine wave is showing the strength (lengths of the arrows) and the direction of electric field at the point on the line of propagation. This point is actually on the line labeled Z. Similarly the red sine wave shows the strength (lengths of the arrows) and the direction of magnetic field at the point on the line of propagation. We know that electromagnetic waves can travel in vacuum, which means there is no medium there. Which means there no actual substance sticking out of the line, in the plane of the sine waves shown below.
An alternate representation could instead use intensity of the color at the points on the line itself to represent the strength of the field. May be two colors each could be used to represent positive and negative strength. Thus we may need four colors - 2 for +ve and -ve electric field and 2 for +ve and -ve magnetic file. But in such representation the picture would simply show a line with varying colored point on the line itself. Thus the electromagnetic wave would be a single dimensional line instead of a 3D shape like in the picture.

So my question is in reality what is the actual shape of the electromagnetic wave? Is it just a line with points, where points on it have sinusoidally varying intensity of electric and magnetic field? Or is the wave sinusoidal shapes like the picture shown below.
What happens to the strength of the electric and magnetic field at the points right next to (close to) the line of propagation, which are not on the line itself? What about all the points on a circle at right angles to the line of propagation. What about the points on that circle that are not the same plane as the sine wave?
Also, it is my understanding that the magnetic lines are always loops - no monopoles, then how can a magnetic filed only have one direction at the point. Would such points not be monopoles?
Hope my questions make sense.
 A: What you are displaying is called an electromagnetic plane wave.  Keep in mind the picture you showed is not the electric and magnetic field at every point in space, it is the field on the points belonging to the line on the z axis. If you were to display the field lines at all points in space it would get very cluttered.

What happens to the strength of the electric and magnetic field at the points right next to (close to) the line of propagation, which are not on the line itself? What about all the points on a circle at right angles to the line of propagation. What about the points on that circle that are not the same plane as the sine wave?

They would all be sine waves that would change as you move the z coordinate and as you go forward/backward in time.

Also, it is my understanding that the magnetic lines are always loops - no monopoles, then how can a magnetic filed only have one direction at the point. Would such points not be monopoles?

Yes, they are monopoles. However, this plane wave solution is not physical - to create it you would need an infinitely long plane of current. In reality you can only have a finite plane and around the edges there would be a 'fringe' effect that would cause the magnetic field lines to loop back on themself.
A: This type of pictures uses the properties of plane waves that all point of a plane perpendicular to the direction of propagation have the same fields vectors at the same instant of time. (That is the reason to be called plane waves). So it is only necessary to represent the field vectors along a line.
Of course the magnitude of the vectors are not lengths, what could be suggested because they are pointing in the $x$ and $y$ direction. There is an implicit mapping $x$  to $E_x$ and $y$ to $B_y$.
That waves are very real locally, to be used for a receiver for example. The magnetic fields are straight lines locally, as the magnetic field of the earth, and its effect in a compass.
A: 
only a representation and not an actual shape right

No. That is actually how they propagate.

no actual substance

The arrows represent the magnitude and direction of the fields at the each point along the line. So focus on a point on the line. There will periodically be electric field and magnetic field at that point. As one diminishes, the other increases. And they are perpendicular.
Try to not make the simple things more complicated than they are.
Our eyes detect these electromagnetic waves.
