Why does harmonic oscillation propagate better? I read that EM radiation can propagate forever only if it follows the harmonic pattern.
If that is true, can you explain why? Why doesn't a different oscillation propagate forever? What happens: is damped, by what? What happens to the energy, is it dispersed in empty space?
Can you also specify which wave is closer to harmonic oscillation among the known ones: produced by the Sun/light bulb/ oscillating circuit/quartz/ laser/ LED etc. ?
 A: There is a serious interpretation difference with what the quote was supposed to convey, and what you understood.

EM radiation can propagate forever only if it follows the harmonic pattern.

The statement is probably referring to the so-called Bandwidth Theorem (see e.g. H. J. Pain, Physics of Vibrations and Waves, Chapter 5), according to which the angular frequency bandwidth of any wave-pulse $\Delta \omega$ and its duration $\Delta t$ are related to another as:
$$\Delta \omega \cdot \Delta t \approx 2 \pi$$
or in terms of linear frequency bandwidth $\Delta \nu$, as:
$$\Delta \nu \cdot \Delta t \approx 1$$
(Similarly, for the spatial part of the wave, the spatial angular frequency spread $\Delta k$ and the spatial extension $\Delta x$ are related to one another as $\Delta k \cdot \Delta x \approx 2 \pi $).
Thus, if we have a wave which has no frequency spread and has a single $\omega$ and $k$, i.e. a pure harmonic wave, of the form $A \cos (\omega t - k x + \phi_0)$, then the duration of the pulse, from the above inequality, would be infinite $(i.e. \Delta \omega = 0 \Rightarrow \Delta t \rightarrow \infty)$. Thus, the wave would last (or propagate) forever.

However, the logically prior question is - Why is the bandwidth theorem at all relevant here? 
(This caters to all but one of your remaining questions:


Why doesn't a different oscillation propagate forever?)

and 

Can you also specify which wave is closer to harmonic oscillation among the known ones: produced by the Sun/light bulb/ oscillating circuit/quartz/ laser/ LED etc.


Actually neither of these real sources emits the kind of idealized harmonic waves one encounters in the textbook, i.e. having the form $A \cos(\omega t - kx)$. In reality you only have discrete wavepulses, each of a finite time-duration, a large number of these put together make up the apparently-continuous stream of light emitted from these real sources. But a very important mathematical artifact which is relevant to this context is - these real wave-pulses of finite duration can be constructed from our textbook-idealized-harmonic waves, which individually would've gone on forever. Formally, this is implemented via Fourier Series and Fourier Transform, wherein one basically superposes a very large number of such ideal waves to construct a real wave-pulse. 
A simpler version, or rather a prototype problem, which illustrates some aspects of the underlying Physics, is that of superposition of a large number ($N$) of oscillations having progressively increasing frequencies (but having the same amplitudes). (cf. H. J. Pain, aforementioned section.) When we analyze the result, one observes that due to unequal frequencies, the component waves gradually develop individual phase difference, and hence start undergoing destructive interference, effectively cancelling each other out. Thus, after a certain duration $\Delta t$, all the component waves have developed sufficient phase differences, such that the resultant has zero amplitude. This time can be roughly interpreted as the approximate "duration" of the pulse. 
Thus, the essence of the bandwidth theorem is that: larger the frequency bandwidth of the superposed waves, ($\Rightarrow$ the sooner the waves would go out of phase), the shorter is the duration of the pulse. This is why single frequency waves (zero bandwidth), last forever.
