  Low energy QCD (below 2 GeV) is a region of resonance dynamics, sometimes
lacking satisfactory description as compared to precision of available
experimental data. Hadronic $\tau$ decays offer a probe for such energy regime.
In general, predictions for decays are model dependent, with parameters fitted
to experimental results. Parameterizations differ by amount of assumptions and
theoretical requirements taken into account. Both model distributions and
acquired data samples used for fits are results of complex effort.
  In this paper, we investigate main parameterizations of $\tau$ decay matrix
elements for the one- and three-prong channels of three-pion $\tau$ decays.
Differences in analytical forms of the currents and resulting distributions
used for comparison with the experimental data are studied. We use invariant
mass spectra of all possible pion pairs and the whole three-pion system. Also
three-dimensional histograms spanned over all distinct squared invariant masses
are used to represent results of models and experimental data.
  We present distributions from {\tt TAUOLA} Monte Carlo generation and
semi-analytical calculation. These are necessary steps in development for
fitting in as model-independent way as possible, and to explore multi-million
event experimental data samples. This includes response of distributions to
model variants, and/or numerical values of parameters. Interference effects of
currents parts are also studied.
  For technical purposes, weighted events are introduced. Even though we focus
on $3\pi\nu_\tau$ modes, technical aspects of our study are relevant for all
$\tau$ decay modes into three hadrons.
