We study the evolution of three-dimensional (3-D), small-scale, small-amplitude perturbations on a plane internal gravity wave using the local stability approach. The plane internal wave is characterised by its non-dimensional amplitude,
$A$
, and the angle the group velocity vector makes with gravity,
$\unicode[STIX]{x1D6F7}$
. For a given
$(A,\unicode[STIX]{x1D6F7})$
, the local stability equations are solved on the periodic fluid particle trajectories to obtain growth rates for all two-dimensional (2-D) and 3-D perturbation wave vectors. For small
$A$
, the local stability approach recovers previous results of 2-D parametric subharmonic instability (PSI) while offering new insights into 3-D PSI. Higher-order triadic resonances, and associated deviations from them, are also observed at small
$A$
. Moreover, for small
$A$
, purely transverse instabilities resulting from parametric resonance are shown to occur at select values of
$\unicode[STIX]{x1D6F7}$
. The possibility of a non-resonant instability mechanism for transverse perturbations at finite
$A$
allows us to derive a heuristic, modified gravitational instability criterion. We then study the extension of small
$A$
to finite
$A$
internal wave instabilities, where we recover and build upon existing knowledge of small-scale, small-amplitude internal wave instabilities. Four distinct regions of the
$(A,\unicode[STIX]{x1D6F7})$
-plane based on the dominant instability modes are identified: 2-D PSI, 3-D oblique, quasi-2-D shear-aligned, and 3-D transverse. Our study demonstrates the local stability approach as a physically insightful and computationally efficient tool, with potentially broad utility for studies that are based on other theoretical approaches and numerical simulations of small-scale instabilities of internal waves in various settings.