Shoulders of NMR peaks
A good question. The term 'shoulder' is really an informal description used to describe the appearance of parts of peaks in the NMR spectrum. When a peak tends to deviate from the standard Lorentzian, we might describe this as having a shoulder. Shoulders can come about from typically three origins:


Shimming anomaly. Fairly common, but the key to diagnosing this cause is that it will affect all peaks in the same way (within the natural linewidth of the peak). So, have a look at the TMS peak, or solvent peak, or that internal reference of ethyl acetate that your supervisor suggested was an impurity. They will all have exactly that same lineshape anomaly, with identical shoulders. Always check these small molecule peaks when you run your spectra to determine shim quality. As an aside on this spectrum, these 'shoulders' you are looking at aren't really shoulders - they are split peaks. If this lineshape was genuinely due to a shimming anomaly, that indicates a pretty serious z shim issue, and you've probably got some floating solid in the sample.
Overlapping peaks. NMR spectra are simply the superposition of all nuclei in the sample. If two singlet species have a non-identical chemical shift within their natural linewidths, then they will appear as a shouldered peak. Obviously, the relative populations will determine how they actually appear. This could account for this type of spectrum (but more than likely doesn't), and that is why you would do other experiments such as COSY or HSQC to verify this.
Splitting due to coupling Often it is difficult to fully resolve small couplings, especially those that come from aromatic systems or long range pathways, where the size of the splitting might be about 1Hz or so. This is where an understanding of the expected coupling pattern for the molecule is very helpful. So, take a deep breath in.....


For Salicylamide, as described in your paper, we would expect H6 to be a large doublet (ortho to H5, ~7-8Hz), but also show meta coupling to H4 (~1-3Hz), and so we expect this large doublet to exhibit small splittings. Indeed, that is what you are seeing. On a good shimming day, in better solvents than dmso, we might even expect to see para splitting (&lt;1Hz - don't hold your breath though). For the H5, we expect, again, a large ortho doublet coupling to H6, but also a large ortho doublet coupling to H4 (again, ~7-8Hz). That gives us the reported triplet, which is in fact a dd. But, we also expect to see some meta coupling to H3 (again 1-3Hz). Again, we see this, and each of these of these should be split due to the small meta couplings. (And we'll leave alone the discussion about whether they have reported apparent splittings, as opposed to J couplings, otherwise J56=J65),  

And breath out.
So what you are seeing is genuine splitting coming from small couplings. These are often not reported in experimental data, because it is quite difficult (or at least time consuming) to resolve all of these couplings. You can get a better understanding of these small splittings if you use resolution enhancement when you process your data. Use a window function such as a small negative line broadened exponential function, or a gaussian function, and you can often split these lines almost to the baseline.