MATLAB: $\tt fft$ and $\tt ifft$ scaling
Whether or not to scale the forward FFT by 1/N depends on which result you want for further analysis: energy (preserving Parseval's identity), or amplitude (measuring height or volts, etc.).  

If you want to measure or analyze energy, then don't scale by 1/N, and a longer sinusoid of the same amplitude will produce a larger FFT result, proportional to the greater energy of a longer signal. 

Slightly more commonly, if you want to measure or analyze amplitudes, then to get a longer sinusoid (thus with more total energy at the exact same amplitude) to produce about the same FFT result as a shorter signal, you will need to scale down the FFT summation by an ratio proportional to the length.  The ratio could be reference_length/N, which is sometimes 1/N if the system input gain is 1.0 for whatever dimensions or units, including the time interval dimensions, that you choose to use in your further analysis.  You need to scale down proportionally because a DFT is a summation: the more you sum up similar items, the bigger the result.

So.  Energy or amplitude.  Which do you want?

Now if you scale down the forward FFT, then you should not scale the inverse so that IFFT(FFT(x)) == x.  Or do vice versa.

The 1/sqrt(N) for scaling seems to me to be for either when one needs a formal symmetry for some proof, or when building some sort of hardware pipeline where the latency and/or number of arithmetic units/gates for the DFT and for the IDFT needs to be identical.  But you get neither a good direct measurement of either energy or amplitude for any typical type of engineering analysis.