Coherence length
The key concept here is the coherence time (or coherence length) of the incoming light. This is the time over which the signal "looks the same" (the maximal shift for which the signal has a strong correlation with its shifted self).
Monochromatic light has infinitely long coherence length : you can shift it by as many periods as you wish, it's still the same sine signal, and therefore as you noted, it can interfere with itself. (Note that the exact position of constructive and destructive interferences is determined by where the path difference is an integer/half-integer number of wavelengths : it depends on the wavelength itself.)
Conversely, white light has a coherence length not much larger than its typical wavelength : you can do interference with it, as long as the path difference is less than or around 1 wavelength.
The coherence length is inversely proportional to the bandwidth of the emitted spectrum. This is easily understood if you remember that the Fourier transform of a gaussian of width $α$ is a gaussian of width $1/α$. Therefore, a light source with a wavelength $λ±δλ$ will have a coherence length of $λ²/δλ$.
Some lasers have coherence lengths of tens of kilometers. Typical cheap laser diodes will be of the order of a few millimeter. White light sources filtered with narrow band-pass filters will be coherent over a few tens of micrometers at best.
So it is entirely possible to do interference with white light, with a few caveats :
* you have to adjust the path difference to within ~ 1 wavelength (i.e. 388
1µm). That's usually extremely inconvenient (read: nervous breakdown…), unless you use a common-path interferometer, where the path difference is ~0 by construction.
* you will see a very limited number of fringes (typically, one…),
* it's better if you filter the light, to increase coherence length. Note that taking a picture of the interferogram with a color camera is equivalent to taking pictures of 3 interferograms filtered with red, blue and green filters ; each of which has about 100nm bandwidth which allows a few fringes. Wikipedia : white light interferometry has a nice example. Note that the fringes will be at different distances for the different colors ; summing them is equivalent to removing the filters, decreasing coherence length.
This has been done many times, most notably by <PERSON> at mount Wilson to measure the diameter of stars.
Beatings
All the above refers to a static (time-average) observation of interference pattern, of a time-delayed version of a signal with itself.
However, when you consider superposition of 2 different signals, even if each of them is monochromatic, the story is fundamentally different. What you get is a time-varying signal whose time-average (over long enough times) is zero.
Consider superposition of 2 monochromatic waves. The addition will be a wave with the average frequency, modulated by the half-difference of the frequencies: $$sin(ω₁t)+sin(ω₂t)=2 sin((ω₁+ω₂)t/2) cos((ω₁-ω₂)t/2)$$ Considering the huge frequencies of visible light, the beating will be too fast to be noticed unless, for a measurement bandwidth Δω :
* the frequencies are extremely close ($ω₁-ω₂<<Δω$),
* both sources are indeed monochromatic (bandwidth $<< Δω$, which pulls in the above discussion of coherence length)
As some lasers have bandwidth in the 10s of kHz and photodetectors can go well into GHz bandwidth, that's actually possible (and called optical heterodyne detection). Note that it's the exact equivalent of some radio recievers, but in the optical frequency range… 795