What is the standard noise channel that is applied in simulations?

Some thoughts:

A theoretical perspective

From a theoretical perspective, the depolarizing channel is the 'standard' (if there is such a thing) or by some means the most applicable.

Because the Paulis (together with the identity operator) form a basis for $SU(2)$, if a code can correct the $X, Y$ and $Z$ flips on a certain qubit (and it it is able to correctly identify no error (i.e. '$I$-flip') having happened), it can correct all errors on that qubit. A theoretical analysis of a code cares a little less about the relative probabilities of these three flips happening, so we just as well can set them all to $frac{p}{3}$, thereby obtaining the depolarizing channel.

The depolarizing channel $Lambda_{mathrm{depo}}$ can be written as:

$$ Lambda_{mathrm{deph}}left(begin{bmatrix}a & b b^{*} & dend{bmatrix}right) = begin{bmatrix} (1-frac{2p}{3})a + frac{2p}{3}d & (1-frac{4p}{3}) b (1-frac{4p}{3})b^{*} & (1-frac{2p}{3})d + frac{2p}{3}aend{bmatrix}. $$ Since $d = 1-a$, we can rewrite this to:

$$ Lambda_{mathrm{deph}}left(rho_{mathrm{in}}right) = (1-frac{4p}{3})rho_{mathrm{in}} + frac{4p}{3} frac{I}{2}, $$ which is a convex combination of the input $rho_{mathrm{in}}$ and the maximally mixed state $frac{I}{2}$. Furthermore, if you would equate $p$ to the elapsed time, $p$ would asymptotically go to $frac{3}{4}$, thereby obtaining the maximally mixed state; therefore the depolarizing channel is in some way the 'worst' noise channel: it destroys both all quantum- (i.e. coherent superpositions) and classical information (there's literally only noise left) in the qubit.

A physical perspective

If you want your simulation to be more true to the physical world, the depolarizing channel is not a very good model, as much as theorists might like it to be. A good first model for noise in qubits is the combination of two channels, the dephasing channel $Lambda_{mathrm{deph}}$ and the amplitude damping channel $Lambda_{mathrm{amp}}$.

Loosely speaking, the dephasing channel map destroys the coherent phase between the $|0rangle$ and $|1rangle$ state:

$$ Lambda_{mathrm{deph}}left(begin{bmatrix}a & b b^{*} & 1-aend{bmatrix}right) = begin{bmatrix}a & e^{-frac{t}{T_{2}}} b e^{-frac{t}{T_{2}}}b^{*} & 1-aend{bmatrix}, $$

where $T_{2}$ is known as the characteristic qubit dephasing time.

The Kraus operators of $Lambda_{mathrm{deph}}$ are $A_{1} = sqrt{1-p}I$ and $A_{2} = sqrt{p}Z$, so it is still a Pauli channel, which can help in the analysis or simulation.

$Lambda_{mathrm{amp}}$ is a little bit trickier: it simulates the relaxation of the excited (by convention $|1rangle$) state, and maps it to the $|0rangle$ state:

$$ Lambda_{mathrm{amp}}left(begin{bmatrix}a & b b^{*} & 1-aend{bmatrix}right) = begin{bmatrix}a & e^{-frac{t}{2T_{1}}} b e^{-frac{t}{2T_{1}}} b^{*} & e^{-frac{t}{T_{1}}}(1-a)end{bmatrix}, $$ where $T_{1}$ is known as the qubit relaxation time.

The Kraus operators are also a little trickier: $B_{1} = begin{bmatrix}1 & 0 0 & sqrt{1-p}end{bmatrix}$ and $B_{2} = begin{bmatrix}0 & sqrt{p} 0 & 0end{bmatrix}$. This means that amplitude damping channel is not a Pauli channel (allthough its Kraus operators can of course be written as linear combinations of the Paulis).

A simulation of a system undergoing both an amplitude damping and dephasing is a good start. There is one big caveat, however: this does not take leakage errors into account whatsoever. Depending on the physical system you are trying to simulate, this might range from either not a very large issue (e.g. for quantum dots) to a gross oversimplification (e.g. for transmon qubits).