Here is a basic question. When does $H^1_{et}(X,\mathbb{Z})$ vanish? Using the exact sequence of constant etale sheaves $0\rightarrow\mathbb{Z}\rightarrow\mathbb{Q}\rightarrow\mathbb{Q}/\mathbb{Z}\rightarrow 0$, it is enough to show that $H^1_{et}(X,\mathbb{Q})$ vanishes. It is known, for instance by 2.1 of Deninger's 1988 JPAA paper, that $H^1_{et}(X,\mathbb{Q})$ vanishes when $X$ is normal.

Note: there are two arguments I think are incorrect that claim to show $H^1_{et}(X,\mathbb{Z})$ always vanishes. The first is that $\mathbb{Z}$ is flasque in the etale topology. This is false. For instance, over the function field $\mathbb{C}(x,y)$, the long exact sequence in cohomology for $0\rightarrow\mathbb{Z}\rightarrow\mathbb{Z}\rightarrow\mathbb{Z}/n\rightarrow 0$ shows that $H^2(\mathbb{C}(x,y),\mathbb{Z})$ is non-zero. So, $\mathbb{Z}$ cannot be flasque. The second argument is that $H^1_{et}(X,\mathbb{Z})=Hom_{cont}(\pi_1^{et}(X),\mathbb{Z})$, where $\pi_1^{et}(X)$ is the etale fundamental group of $X$, which is a profinite group. Since it is profinite, the $Hom$ group above vanishes. But, the claimed equality between $H^1_{et}(X,-)$ and $Hom_{cont}(\pi_1^{et}(X),-)$ only holds for torsion sheaves, as far as I have been able to determine.

I am in fact interested in several things. First, either an example of $X$ such that $H^1_{et}(X,\mathbb{Z})$ is non-zero, or a proof that this always vanishes. Second, the same thing but where we only look at affine $X$. In particular, if it exists, I would love to see an example of a commutative ring $R$ where $H^1_{et}(Spec R,\mathbb{Z})$ is non-zero, if this is possible.