  We study the Bianchi-I cosmological model motivated by signals of statistical
isotropy violation seen in cosmic microwave background (CMB) observations and
others. To that end, we consider various kinds of anisotropic matter that
source anisotropy in our model, specifically Cosmic strings, Magnetic fields,
Domain walls and Lorentz violation generated magnetic fields. These anisotropic
matter sources, taking one at a time, are studied for their co-evolution with
standard model (isotropic) sources viz., dust-like (dark/normal) matter, and
dark energy modelled as cosmological constant. We constrain the Hubble
parameter, density fractions of anisotropic matter, cold dark matter (CDM), and
dark energy ($\Lambda$) in a Bianchi-I universe with planar symmetry i.e.,
which has a global ellipsoidal geometry, and try to find signatures of a cosmic
preferred axis if any. The latest compilation of Type Ia Supernova (SNIa) data
from Pantheon+SH0ES collaboration is used in our analysis to obtain constraints
on cosmological parameters and any preferred axis for our universe. In our
analysis, we found mild evidence for a cosmic preferred axis. It is interesting
to note that this preferred axis lies broadly in the vicinity of other
prominent cosmic anisotropy axes reported in the literature from diverse data
sets. Also we find some evidence for non-zero (negative) cosmic shear and
eccentricity that characterize different expansion rates in different
directions and deviation from an isotropic scale factor respectively. The
energy density fractions of two of the sources considered are found to be
non-zero at a $2\sigma$ confidence level. To be more conclusive, we require
more SNIa host galaxy data for tighter constraints on distance and absolute
magnitude calibration which are expected to be available from the future JWST
observations and others.
