  Cosmology faces a pressing challenge with the Hubble constant ($H_0$)
tension, where the locally measured rate of the Universe's expansion does not
align with predictions from the cosmic microwave background (CMB) calibrated
with $\Lambda$CDM model. Simultaneously, there is a growing tension involving
the weighted amplitude of matter fluctuations, known as $S_{8,0}$ tension.
Resolving both tensions within one framework would boost confidence in any one
particular model. In this work, we analyse constraints in $f(T)$ gravity, a
framework that shows promise in shedding light on cosmic evolution. We
thoroughly examine prominent $f(T)$ gravity models using a combination of data
sources, including Pantheon+ (SN), cosmic chronometers (CC), baryonic acoustic
oscillations (BAO) and redshift space distortion (RSD) data. We use these
models to derive a spectrum of $H_0$ and $S_{8,0}$ values, aiming to gauge
their ability to provide insights into, and potentially address, the challenges
posed by the $H_0$ and $S_{8,0}$ tensions.
