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The ``Spectroscopy and H-band Imaging of Virgo cluster galaxies'' (SHIVir) survey is an optical and near-infrared survey which combines SDSS photometry, deep H-band photometry, and long-slit optical spectroscopy for 190 Virgo cluster galaxies (VCGs) covering all morphological types over the stellar mass range log (M_*/M_Sun) = 7.8-11.5$. We present the spectroscopic sample selection, data reduction, and analysis for this SHIVir sample. We have used and optimised the \texttt{pPXF} routine to extract stellar kinematics from our data. Ultimately, resolved kinematic profiles (rotation curves and velocity dispersion profiles) are available for 133 SHIVir galaxies. A comprehensive database of photometric and kinematic parameters for the SHIVir sample is presented with: grizH magnitudes, effective surface brightnesses, effective and isophotal radii, rotational velocities, velocity dispersions, and stellar and dynamical masses. Parameter distributions highlight some bimodal distributions and possible sample biases. A qualitative study of resolved extended velocity dispersion profiles suggests a link between the so-called ``sigma-drop'' kinematic profile and the presence of rings in lenticular S0 galaxies. Rising dispersion profiles are linked to early-type spirals or dwarf ellipticals for which a rotational component is significant, whereas peaked profiles are tied to featureless giant ellipticals.