This is the status as far as I know. For dimension $\leq 2$ it is up to date. For higher dimensional representations I'm sure it is very incomplete, so feel free to edit or comment.

**Dimension 1.** Known by "Artin-Hecke".

**Dimension 2.** Only open case is even $A_5$ representations. References for the known cases are:

$C_n$ "Artin-Hecke"

$D_n$ "Artin-Hecke" (see here)

$A_4$ Langlands, "Base change for GL(2)" (1980)

$S_4$ Tunnell, "Artin's conjecture for representations of octahedral type" (1981)

$A_5$ (odd $\rho$ over $\mathbb{Q}$) Khare-Wintenberger, "Serre's modularity conjecture (I)" (2009)

$A_5$ (odd $\rho$ over totally real fields) Pilloni-Stroh, "Surconvergence, ramification et modularité" (2013)

**Dimension 3.** Mostly wide open.

($\rho$ induced) Jacquet, Piatetski-Shapiro, Shalika, "Relèvement cubique non normal" (1981)

($\rho$ twist of a symmetric square) Gelbart, Jacquet, "A relation between automorphic representations of GL(2) and GL(3)" (1978)

**Dimension 4.** Only open solvable cases are $E_{2^4}\cdot D_{10}$ and $E_{2^4}\cdot F_{20}$. There are known non-solvable cases, but in general it's wide open.

($\rho$ solvable) $\mathrm{GO}_4$ Ramakrishnan, "Modularity of solvable Artin representations of GO(4)-type" (2001)

($\rho$ solvable) $E_{2^4}\cdot C_5$ Martin, "A symplectic case of Artin's conjecture" (2003)

**Dimension $\geq 5$** Again there are some known cases, but mostly wide open.

The consensus is that a solution of the complete Artin conjecture is only accesible from general functoriality results such as base change or induction that would imply the strong Artin conjecture, and therefore Artin's holomorphy conjecture.

This means that the only way we have to prove that an Artin L-function $L(\rho, s)$ is holomorphic is to prove that $\rho$ is modular.

An alternative, as pointed out by Julien in the comments, would be to solve Selberg's orthogonality conjecture. That is implies the Artin conjecture is proved in "Selberg's Conjectures and Artin L-functions" (1994).