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I would like to know if hydrogens that are not bound to a carbon (such as in groups $\ce{-OH}$ and $\ce{-NH2}$) would lead to the presence of a singlet peaks in the NMR spectra or not, and if we have to count or omit them in the integration of the spectrum. I thought I was sure about it but lately I noticed that it depends on the spectra.

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Yes, with a but.

First the yes: they do appear, albeit at slightly different chemical shifts as, of course, the environment around a proton in $\ce{-CH2\!\;\!-}$ is slightly different to the electronic environment around the proton in $\ce{-NH2}$. The $\ce{N-H}$, $\ce{O-H}$, and $\ce{C-H}$ bonds all have slightly different electronegativies.

Now for the but: the thing that more significantly distinguishes $\ce{N-H}$ and $\ce{O-H}$ from $\ce{C-H}$ is their lability in protic solvents. Exchange of labile protons in the sample molecule with deuterons (i.e. deuterium ions, $\ce{^2H+}$) in the solvent can can remove the proton peak from the spectrum (this exchange is an equilibrium process, but as the solvent is in large excess the equilibrium lies over towards full deuteration of the sample's labile groups. This can actually be useful in identifying peaks: first run a sample in, for example, $\ce{CDCl3}$, then add a couple of drops of $\ce{D2O}$ to your NMR tube, give it a shake, and re-run the spectrum -- your labile protons will have vanished.

Edit 1: Furthermore, as @user1573870 points out in the comments, exchange of protons (exhanging protons for protons, rather than exchanging protons for deuterons), on "the NMR timescale" can lead to broadening, and the resulting peak can be, while technically present, very difficult to spot. I'd suggest anyone doing NMR do background reading on that, because it's both useful and interesting (and I need to do some, because I'm out of practice with NMR).

Edit 2:To directly answer @Chewie's question whether "we have to count or omit them in the integration of the spectrum", you only count each of the peaks that are present. Using the example of ethanol, as asked in the comments, $\ce{CH3CH2OH}$ should give three peaks:

  1. The $\ce{CH3}-$ peak should integrate to show three protons,
  2. The $\ce{-CH2}-$ peak should integrate to show two protons, and
  3. The $\ce{-OH}$ peak should integrate to show one proton.

If due to exchange with the deuterated solvent the $\ce{-OH}$ peak is not present, or if the $\ce{-OH}$ peak is 'there' but so broad as to be invisible to you (or so broad that it is difficult to integrate accurately), then you would only give the integrals for the other two peaks, and those two peaks should give integrals with a ratio equivalent to their respective number of protons, i.e. 3:2. If the $\ce{-OH}$ peak is not there, its proton does not just get thrown in with one of the other peaks/integrals (unless, sneaky proton, it just happened to have the same integral as another peak, and was hidden behind it, but for something as simple as $\ce{CH3CH2OH}$ that is unlikely).

The punchiest way to say all of thise would be that one does not "integrate spectra"; one integrates peaks.

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    $\begingroup$ Also, such labile protons (e.g. acidic ones) do often give rise to a broadened peak due to having exchange rates on the NMR time scale. In extreme cases, this can broaden the peak to an extent that you might even overlook it. $\endgroup$ – user1573870 Jan 23 '17 at 16:08
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    $\begingroup$ @Chewie Sorry that I didn't address that explicitly. I'll add that in. $\endgroup$ – owjburnham Jan 23 '17 at 16:36
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    $\begingroup$ Even if exchange is slow on the nmr timescale, proton peaks may be broadened due to quadrupolar relaxation. This is often the case with $\ce{N-H}$ protons. See here for some background. $\endgroup$ – ron Jan 23 '17 at 17:09
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    $\begingroup$ @Chewie Ethanal and ethanol would each have a $\ce{-CH3}$ peak, albeit in different places due to different environments; however, the other peak in each spectrum would be radically different: in the case of ethanol, an alkane $\ce{-CH2}$ around 4 ppm, shifted 'upfield' by proximity to the $\ce{OH}$, and in the case of ethanal (acetaldehyde) an aldehyde $\ce{C-H}$ at around 10ppm. Furthermore, the ethanol $\ce{-CH2}$ would integrate to 2, while the ethanal $\ce{C-H}$ would integrate to 1. $\endgroup$ – owjburnham Jan 23 '17 at 17:52
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    $\begingroup$ Peaks from exchangeable protons do not just "vanish". They reappear as a shifted water (HDO) peak, at a position between the original and a native water peak in the same solvent. And it all depends on the relative molar concentrations of sample and water. $\endgroup$ – Karl Jan 23 '17 at 18:23

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