$A^{\alpha}=(\phi,\vec{A})$ or $A_{\alpha}=(\phi,\vec{A})$? I have seen both equations, $A^{\alpha}=(\phi,\vec{A})$ is from Wikipedia and $A_{\alpha}=(\phi,\vec{A})$ is from my lecture. Which one is right?
My thoughts: As far as I know, $A$ is a 1-form, so $A(p)\in T_p^*M$ for all $p\in M\subset\mathbb R^4$. In addition, $(\text{d}x^1{}_p,...,\text{d}x^4{}_p)$ with $(x^1,...,x^4):=\text{id}_M$ is a basis of $T_p^*M$ and since we normally write the coefficients of dual vectors with the index below, I'd say $A=A_{\alpha}\text{d}x^{\alpha}$.
 A: As the existing answer says, it's $A^\mu = (\phi, \mathbf{A})$. Here's a simple way to see that. In Lorenz gauge, the equation of motion is
$$\partial^2 A^\mu = J^\mu.$$
We also know that $J^\mu = (\rho, \mathbf{J})$, and that the components of this equation are
$$\partial^2 \phi = \rho, \quad \partial^2 \mathbf{A} = \mathbf{J}.$$
There are no minus signs anywhere, so we must have $A^\mu = (\phi, \mathbf{A})$.
A: It's $A^\mu = (\phi, \mathbf{A})$, no matter your metric convention.
Proof:
I can never remember the signs in the correspondence between $F_{\mu\nu}$ and the electric and magnetic field (especially since they depend on the metric signature), so instead let's look at the Lorentz force per unit charge
$$f^\mu = F^\mu{}_\nu u^\nu.$$
The part of $f^i$ proportional to $u^0$ will then be the electric field $E^i$:
$$f^i = F^i{}_\mu u^\mu = F^i{}_0 u^0 + \dots = (\partial^i A_0 - \partial_0 A^i) u^0 + \dots.$$
Now, no matter the metric signature we have $\partial^i A_0 = -\partial_i A^0$, because we're flipping one time index and one space index, so we arrive at
$$E^i = -\partial_i A^0 - \partial_0 A^i.$$
Comparting with the known formula $\mathbf{E} = - \nabla \phi - \partial \mathbf{A}/\partial t$, we see that the contravariant components of $A^\mu$ are the potentials with no sign change.
