The ring $A = \prod_{n=1}^{\infty} \mathbb{F}_2$ has some interesting/disturbing properties.
For example, the affine scheme $X := {\rm{Spec}}(A)$ has non-open connected components (since it has infinitely many open points), all local rings on $X$ are noetherian (in fact they're all $\mathbb{F}_2$ since $a^2 = a$ for all elements $a$) even though $A$ is not noetherian, and if $I$ is an ideal that isn't finitely generated then ${\rm{Spec}}(A/I) \hookrightarrow X$ is formally unramified (since closed immersion), finite type, and flat but not etale étale (since not finitely presented) and not open, in contrast with the noetherian case.
The ring $A = \prod_{n=1}^{\infty} \mathbb{F}_2$ has some interesting/disturbing properties. For example, the affine scheme $X := {\rm{Spec}}(A)$ has non-open connected components (since it has infinitely many open points), all local rings on $X$ are noetherian (in fact they're all $\mathbb{F}_2$ since $a^2 = a$ for all elements $a$) even though $A$ is not noetherian, and if $I$ is an ideal that isn't finitely generated then ${\rm{Spec}}(A/I) \hookrightarrow X$ is formally unramified (since closed immersion), finite type, and flat but not etale (since not finitely presented) and not open, in contrast with the noetherian case.