An automorphic L-function $L(\pi,s)$ belongs to the Selberg class if and only if the generalized Ramanujan conjecture for $\pi$ is known.

This includes two important cases: Hecke characters and holomorphic modular forms.

Automorphicity of Rankin-Selberg convolution is known in that range for $\mathrm{GL}(1)\times \mathrm{GL}(1)$ and $\mathrm{GL}(1)\times \mathrm{GL}(2)$ (classical) and $\mathrm{GL}(2)\times \mathrm{GL}(2)$ (Ramakrishnan).

Therefore the answer includes Hecke L-functions and L-functions of modular forms.

I'm not sure if there are any more cases where both neccesary results are known. For example for Maass forms the Ramanujan conjecture is missing, and I don't think much is known in general about the modularity for higher $n$. $\mathrm{GL}(2)\times \mathrm{GL}(3)$ alredy seems to be open, see this answer by Paul Garrett.

An automorphic L-function $L(\pi,s)$ belongs to the Selberg class if and only if the generalized Ramanujan conjecture for $\pi$ is known.

This includes two important cases: Hecke characters and holomorphic modular forms.

Automorphicity of Rankin-Selberg convolution is known in that range for $\mathrm{GL}(1)\times \mathrm{GL}(1)$ and $\mathrm{GL}(1)\times \mathrm{GL}(2)$ (classical) and $\mathrm{GL}(2)\times \mathrm{GL}(2)$ (Ramakrishnan).

Therefore the answer includes Hecke L-functions and L-functions of modular forms.

I'm not sure if there are any more cases where both neccesary results are known. For example for Maass forms the Ramanujan conjecture is missing. See the comments for more information.

An automorphic L-function $L(\pi,s)$ belongs to the Selberg class if and only if the generalized Ramanujan conjecture for $\pi$ is known.

This leaves only two possibilities: Hecke characters and holomorphic modular forms.

Automorphicity of Rankin-Selberg convolution is known in that range for $\mathrm{GL}(1)\times \mathrm{GL}(1)$ and $\mathrm{GL}(1)\times \mathrm{GL}(2)$ (classical) and $\mathrm{GL}(2)\times \mathrm{GL}(2)$ (Ramakrishnan).

Therefore, if I've understood the question correctly, the answer is: Hecke L-functions and L-functions of modular forms.

An automorphic L-function $L(\pi,s)$ belongs to the Selberg class if and only if the generalized Ramanujan conjecture for $\pi$ is known.

This includes two important cases: Hecke characters and holomorphic modular forms.

Automorphicity of Rankin-Selberg convolution is known in that range for $\mathrm{GL}(1)\times \mathrm{GL}(1)$ and $\mathrm{GL}(1)\times \mathrm{GL}(2)$ (classical) and $\mathrm{GL}(2)\times \mathrm{GL}(2)$ (Ramakrishnan).

Therefore the answer includes Hecke L-functions and L-functions of modular forms.

I'm not sure if there are any more cases where both neccesary results are known. For example for Maass forms the Ramanujan conjecture is missing, and I don't think much is known in general about the modularity for higher $n$. $\mathrm{GL}(2)\times \mathrm{GL}(3)$ alredy seems to be open, see this answer by Paul Garrett.