There is a nice review article Low-barrier hydrogen bonds in proteins
by M.V. Hosur, R. Chitra, Samarth Hegde, R.R. Choudhury, Amit Das, and R.V. Hosur
Most hydrogen bonds in proteins are weak, as characterised by a donor-acceptor distance larger than 2.8 Angstroms, and interaction energies of a few kcal/mol (~0.1 eV~3 k_B T). However, there are some bonds that are much shorter. In particular, Cleland proposed in 1993 that for some enzymes that there are H-bonds that are sufficiently short (R ~ 2.4-2.5 A) that the energy barrier for proton transfer from the donor to acceptor is sufficiently small that it is comparable to the zero-point energy for the donor-H stretch vibration. These are called low-barrier hydrogen bonds. This proposal remains controversial. For example, Ariel Warshel says they have no functional role.
The authors perform extensive analysis of crystal structure databases, for both proteins and small molecules, in order to identify the relative abundance of short bonds, and their location relative to the active sites of proteins. Here are a few things I found interesting.
1. For a strong bond, the zero-point motion along the bond direction will be much larger than in the perpendicular directions. This means that there should be significant anisotropy in the ellipsoid associated with the uncertainty of the hydrogen atom position determined from neutron scattering [ADP = Atomic Displacement Parameter = Debye-Waller factor]. The ellipsoid is generally spherical for normal [i.e., common and weak] H-bonds. They find that anisotropy is correlated with the presence of short bonds and with "matching pK_a's" [i.e., the donor and acceptor have similar chemical identity and proton affinity], as one would expect.
2. For 36 different protein structures they find very few LBHB's. Furthermore, in many the H-bonds identified are away from the active site. But, this may be of significance, as discussed below.
3. A LBHB may play a role in excited state proton transfer in green fluorescent protein, as described here.
4. In HIV-1 protease there is a very short H-bond with no barrier.
5. There is correlation between the location of short H-bonds and the "folding cores" of specific proteins, including HIV-1. These sites are identified through NMR, that allows one to study partially denatured [i.e. unfolded] protein conformations. This suggests short H-bonds may play a functional role in protein folding.