There are a few examples that are finitely generated.
(1) Let $L$ be Lyndon's groupoid given by the following multiplication table: \begin{array} [c]{c|ccccccc} L & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 4 & 0 & 4 & 5 & 6 & 0 & 0 & 0 \\ 5 & 0 & 5 & 5 & 5 & 0 & 0 & 0 \\ 6 & 0 & 6 & 6 & 6 & 0 & 0 & 0 \\ \end{array} Then the variety $\mathrm{var}\, L$ is non-finitely based and an explicit basis is: \begin{align} (xx)y = x(yz) = zz, \quad (\cdots((xy_1) y_2) \cdots) y_k = ((\cdots((xy_1) y_2) \cdots) y_k) y_1, \\ ((\cdots(x_1 x_2) \cdots )x_k )x_1 = zz, \quad k=1,2,\ldots \end{align}
(2) Let $A_2$ be the 0-simple semigroup $$ \langle a,b \mid a^2=aba=a,\ bab=b,\ b^2=0\rangle $$ of order five and let $\mathbb{Z}_n$ be the cyclic group of order $n$. Then for each $n \geq 2$, the variety $\mathrm{var} \{A_2,\mathbb{Z}_n\}$ is non-finitely based and an explicit basis is: \begin{align} (xy)z=x(yz), \quad x^2 = x^{n + 2}, \quad xyx = x (yx)^{n + 1}, \quad xyxzx = xzxyx, \\ (x_1^n x_2^n \cdots x_k^n)^{3} = (x_1^n x_2^n \cdots x_k^n)^{2}, \quad k=2,3,\ldots \end{align}
As for non-finitely generated varieties, apart from the example of Perkins (1969), there is an easy to describe example by J. R. Isbell (1970): the variety of monoids defined by $$ (x^py^p)^2 = (y^px^p)^2, \quad p = 2,3,5,7,11,\ldots \text{(primes)} $$