The reactivity of NHS activated esters has mostly been desribed quite anecdotically. They are susceptible to hydrolysis in water, the kinetics of which is dependent on the nature of the NHS activated group (ester vs carnonate vs carbamate) as well as on the pH of the reaction buffer1, which is one way to show that primary amines are not the only nucleophile which can react with NHS-activated esters.
As for the chemical groups that have been observed to react with NHS activated esters, I did find a couple of examples upon looking up this issue. For instance, the Zenobi group have attempted a quite systematic study on the groups that can react with doubly NHS activated cross-linkers. Citing the abstract2 :
As soon as additional cross-linkers were attached or loops were formed, other amino acids were also involved in the reaction. In addition to the primary amino groups, serine, threonine and tyrosine showed significant reactivity due to the effect of neighboring amino acids by intermediate or permanent Type-1 cross-link formation. The reactivity is highly dependent on the pH and on adjacent amino acids.
In an earlier work, the reactivity of a biotinylation NHS-ester based reagent was assessed on a single peptide, [D-Lys(6)]gonadotropin releasing hormone. A similar reactivity of -OH groups was observed, with in addition that of the arginine guanidinium group3:
In addition to the O-acylation of Ser(4) and Tyr(5) in this peptide, we have also identified a novel biotinylation of the Arg(8) side chain.
Other groups have also observed some reactivity with cysteins (sulfhydryl groups), but in a situation where the reagent is included within a ligand of the target protein. One can assume, as do the authors, that specific positionning of the ligand within the binding site can lead to binding with a cystein group4,5.
From these earlier work, one cas conclude, in answer to your question that although primary amines are the most reactive group for NHS-ester derivatization, other groups present in peptide side-chains (-OH for tyrosine, serine, threonine, guanidinium for arginine and sulfhydryls for cysteins).
The last issue raised in the question remains that of the secondary amine groups. It is widely accepted that pH influences strongly the kinetics of NHS derivatization, mostly through protonation of the amine groups which cannot act as nucleophiles in protonated state. This has been put to use in controlling the buffer pH in order to favor N-terminal reaction vs lysine side chain reaction. Although, in my experience, it is not at all a straightforward experiment and it is highly dependent on the nature of the derivatization group. One could hypothesize that since secondary amines are more basic then primary amines, the reactivity of secondary amines towards NHS should be reduced compared to primary amines, but I have no experimental or litterature proof for this part of the answer.