When thinking about this question I thought about degree $p$ extensions of a field $F$ of $\text{char}(F)=p>0$.
In general we distinguish finite extensions according to the properties normal and separable.
We know that purely inseparable extensions are normal. Although there are non-normal inseparable extensions they cannot be of degree $p$, because the inseparable part "takes up all of the degree p" and thus an inseparable degree $p$ extension must be purely inseparable. Such extensions are given by an irreducible polynomial of the form $X^p-\alpha$ for $\alpha\in F$. This classifies these extensions up to the question for which $\alpha\in F$ the polynomial $X^p-\alpha$ is irreducible.
According to Artin-Schreier theory all normal separable degree $p$ extensions (i.e. Galois extensions and in this case in particular cyclic extensions) are given by an irreducible polynomial of the form $X^p-X-\alpha$ for $\alpha\in F$. This classifies these extensions up to the question for which $\alpha\in F$ the polynomial $X^p-X-\alpha$ is irreducible.
The remaining class are non-normal separable degree $p$ extensions. Playing around with Magma I found such an extension over $F=\mathbb{F}_p(t)$ given by $f=X^p-tX-1$ for $p=3$ - see code below: Magma tells us that the defining polynomial should have an order $6$ Galois group. Jyrki Lahtonen eluminates this by an explanation in the comments showing that this extension is indeed non-normal separable of degree $p$. Experimentially this type of polynomial seems to work for all primes $2\neq p\leq 41$ where I stopped searching because of the time restrictions of the free Magma calculator.
p := 3;
E<t> := FunctionField(GF(p));
R<X> := PolynomialRing(E);
F<a> := FunctionField(X^p-t*X-1);
F;
Characteristic(F);
Degree(F);
f := DefiningPolynomial(F);
G := GaloisGroup(f);
Order(G);
My question is: Is there a nice classification for non-normal separable degree $p$ extensions? Is such a classification parallel to the other two cases, i.e. are such extensions given by polynomials of a certain standard form?
Some results:
Any extension of a finite field is always normal and separable for arbitrary degree. Even better, for every fixed degree $n$ there is a unique degree $n$ extension up to isomorphism. Thus degree $p$ extension can easily be classified: there is exactly one such extension and it is of Artin-Schreier type.
Any extension of degree $2$ is necessarily normal. So in case of characteristic $2$ any extension is either of Artin-Schreier type or purely inseparable. This seems to be one of the few cases where the "$2$ is the oddest prime"-exceptionalism works in favor of simplifying the question instead of complicating it.
Remarks:
Forming the composite field of a non-normal separable $p$-extension (as above) with a purely inseparable $p^{n-1}$-extension we get non-normal inseparable extension of degree $p^n$, so for any $p$-power. The example linked above of a non-normal inseparable extension has degree $15$ and is thus not a $p$-power.
Forming the composite field of several non-normal separable $p$-extensions defined by $n$ suitable extensions one should get a non-normal inseparable extension of degree $p^n$, so for any $p$-power. Suitable here means that one has to avoid configurations analog to $\mathbb{Q}(\sqrt[3]{2},\sqrt[3]{3},\sqrt[3]{6})/\mathbb{Q}$, which will not be an extension of degree $3^3$. I don't have a proof that such extensions really exist for every $p^n$, i.e. that there are enough suitable non-normal separable $p$-extensions, neither in general nor for an example field.
These remarks show how an answer to the question is useful to a more general study of field extensions.
Update: The question was rewritten in light of some computer experiments and comments on that.