  Tailored topological stabilizer codes in two dimensions have been shown to
exhibit high storage threshold error rates and improved subthreshold
performance under biased Pauli noise. Three-dimensional (3D) topological codes
can allow for several advantages including a transversal implementation of
non-Clifford logical gates, single-shot decoding strategies, parallelized
decoding in the case of fracton codes as well as construction of fractal
lattice codes. Motivated by this, we tailor 3D topological codes for enhanced
storage performance under biased Pauli noise. We present Clifford deformations
of various 3D topological codes, such that they exhibit a threshold error rate
of $50\%$ under infinitely biased Pauli noise. Our examples include the 3D
surface code on the cubic lattice, the 3D surface code on a checkerboard
lattice that lends itself to a subsystem code with a single-shot decoder, the
3D color code, as well as fracton models such as the X-cube model, the
Sierpinski model and the Haah code. We use the belief propagation with ordered
statistics decoder (BP-OSD) to study threshold error rates at finite bias. We
also present a rotated layout for the 3D surface code, which uses roughly half
the number of physical qubits for the same code distance under appropriate
boundary conditions. Imposing coprime periodic dimensions on this rotated
layout leads to logical operators of weight $O(n)$ at infinite bias and a
corresponding $\exp[-O(n)]$ subthreshold scaling of the logical failure rate,
where $n$ is the number of physical qubits in the code. Even though this
scaling is unstable due to the existence of logical representations with $O(1)$
low-rate Pauli errors, the number of such representations scales only
polynomially for the Clifford-deformed code, leading to an enhanced effective
distance.
