Q: Spectrum of LEDs How monochromatic is the light from LEDs? I want a set of light sources which will emit light of different wavelengths, as narrow a spectrum as practical.
How wide are the spectrums of typical LEDs?

A: You will be getting a narrow band, many LEDs have width of about 20 nm at 50% down from the peak; other LEDs have quite a wide band.  The datasheets for LEDs will give you a curve for those specific to a particular LED.
Here are the graphs of LEDs I chose for a scientific imaging project based on how narrow their bands were without resorting to exotic components. I wanted a selection of colours as narrow as possible (but cheap), and bought well-known manufacturers' parts with datasheets, which I summarised as follows:

(The 400 nm line is dotted because it was an estimate.  The others were traced from datasheets and normalised to give same height.)
If you need narrower:


*

*a laser will be better than 1 nm, and can be much much better. (Given as 0.85 in datasheet of cheapest laser at a distributor). Some "broadband" lasers have a linewidth of a few to about 10 nm (Wikipedia)

*some extrenely narrow bandpass filters are available, such as astronomical  sodium filter, which can have a typical bandpass of 0.05 nm.  Be warned they can be extremely expensive.


A: 
I want a set of light sources which will emit light of different wavelengths, as narrow a spectrum as practical.

If you want as narrow as practical, you should use lasers. LEDs are intrinsically broadband light sources. 
However you really should define a bandwidth you need. Too narrow will introduce problems such as speckle.

A: Different WAVELENGTHS may be difficult.  Different COLOURS, achieved by mixing the output of three LEDs, red, blue and green, is easy, and cheaply available in a single package.
https://www.rapidonline.com/truopto-ostama5b62a-full-colour-5mm-led-60-diffused-72-9647?utm_source=AffWin&utm_medium=Affiliate&awc=1799_1579713939_2a7737a0902bbca9c2076b48d027e40a
