Why is a filament lamp an incoherent light source? I've tried to search for this answer but found no proper explanation.
My previous ideas were...
1) It is not a point source, so the different points on a filament lamp may cancel out the light from other points on a filament lamp
2) It emits light rays at random (even though I've read this, I don't exactly understand what it means. Does it mean a single point on the filament may emit a photon one second and then the next second not emit anything?)
Are these reasons along the right lines? Could someone give a proper explanation as to why a filament lamp is an incoherent light source?
 A: There exists two types of coherences:


*

*temporal coherence, 
which describes how the field amplitude $u$ at time $t$ is correlated with the field amplitude at time $t+\tau$, 
$$\langle u(x,t)\cdot u(x,t+\tau) \rangle
= \lim_{T\to \infty} \frac{1}{T} \int_{-T/2}^{T/2} u(x,t)\cdot u(x,t+\tau) \,dt
$$
When considering temporal coherence, we are interested in the ability of a light beam to interfere with a temporally delayed (but not spatially shifted) version of itself. Hence, we usually assume that the light source is a point object.  
Thermal light sources emit blackbody radiation with a spectral distribution given by Planck's law. The coherence time can be estimated as 
$$
\tau_c = \frac{\hbar}{k_B T}
$$
So, increasing the temperature of the filament decreases the coherence time. For a temperature around  $3000K$ we obtain a coherence time on the order of $10^{-15}s$, which corresponds to a coherence length $c_0\tau_c \approx 1\mu m$. Hence, filament light sources do not possess a "great" temporal coherence. However, this is not the main obstacle to use thermal light sources in coherence experiments.

*spatial coherence, where we drop the assumption of an ideal point source and take the spatial extension of the light source into account. When considering spatial coherence we are interested in the ability of a light beam to interfere with a spatially shifted version of itself.  
The spatial coherence can be tested by using Young's double slit experiment. If the two slits have a distance $d$ between them,   we are analysing the correlation 
$\langle u(x-d/2, t) \cdot u(x+d/2, t+\tau)\rangle$. Note that we also take the time difference due to the two different path length into account: Considering the two paths from the source to the point $P$, the path passing through the upper slit is shorter.

The  spatial coherence of blackbody radiation is approx. given by $\sin(k d)/(kd)$, where $k=2\pi / \lambda$ is the wave number. Hence, the spatial coherence is on the order of the wavelength, while the light emitting filament has a much larger surface. Thus, in order to obtain a spatially coherent light source, we have to reduce the size of the filament, which contributes to the experiment -- e.g. by using an aperture. 
Thus, for visible light we have to use an aperture in the micrometer range. Most of the light emitted by the filament would be lost at the aperture. The thermal source becomes extremely inefficient.
A: The most significant recent development in the double slit experiment was running the experiment with single photons (1960s - you can google it).  What's remarkable is one still gets the "interference" pattern ... but it seems impossible, how did a single photon manage to "interfere".  You will read statements like the photon interferes with itself, or the the photon goes thru both slits, in physicists minds the statements are necessary because "interference" is/was the only logical theory.  And of course to get interference you must have coherency in the source as you have noticed.  But we can resolve the single photon experiments by employing Feynman's theory on the path integral of light, photons are travelling to the screen in probable paths (n times wavelength). This happens whether the source is mainly incoherent or coherent, whether we have single photon source or a high intense one.
So what does coherency mean ... its becomes a complex measure of correlation of photons (see Semoi's answer) .... more simply it boils down to how similarly the photons from a source are behaving, a laser has many similar photons, a light bulb much less ( but still some level of coherence/correlation). The more coherent the more clearly visible the "interference" pattern.  
A: Because filament lamp isn't just a single source, but consists of enormous numbers of different sources. Different atoms emits light in random manner. There is no particular pattern in emission of light from these sources. And that's why at some point, the phase difference of different light waves are not constant. That's why light emitting from filament lamp is incoherent.
A: You are partly right but it also comes down to the different energy levels of the electrons. It is a random process as to when  electrons will fall to lower energy levels releasing their energy. The direction that energy is released is also random.
