Is there a list of common finite semigroups? I'm assuming a sensible classification as one has with finite simple groups is impossible, but it would be nice to see a list of common small examples, however incomplete.
For example, if one is testing a simple proposition about finite semigroups against some examples, what does one start with?
Common semigroups of low order which are not groups. The semigroup $N_2 = \{a,0\}$ where $0$ is a zero and $a^2 = 0$. The monoid $U_1 = \{1, 0\}$ under usual multiplication of integers.
Let $I$ and $J$ be two nonempty sets. Define an operation on $I \times J$ by setting, for every $(i,j), (i',j') \in I \times J$, $$ (i,j)(i',j') = (i, j') $$ This defines a semigroup of order $|I||J|$, usually denoted by $B(I, J)$.
Second part of your question. Let $B_2$ be the set of $2 \times 2$ matrices with $0$-$1$ entries and at most one nonzero entry. Equipped with the usual multiplication of matrices, $B_2$ is a semigroup. $$ B_2 = \left\{ \begin{pmatrix} 1&0 \\ 0&0 \end{pmatrix}, \begin{pmatrix} 0&1 \\ 0&0 \end{pmatrix}, \begin{pmatrix} 0&0 \\ 1&0 \end{pmatrix}, \begin{pmatrix} 0&0 \\ 0&1 \end{pmatrix}, \begin{pmatrix} 0&0 \\ 0&0 \end{pmatrix} \right\} $$ Setting $a=\left( \begin{smallmatrix} 0\ 1\\ 0\ 0 \end{smallmatrix} \right) $ and $ b=\left( \begin{smallmatrix} 0\ 0\\ 1\ 0 \end{smallmatrix} \right) $, one gets $ab=\left( \begin{smallmatrix} 1\ 0\\ 0\ 0 \end{smallmatrix} \right) $, $ba=\left( \begin{smallmatrix} 0\ 0\\ 0\ 1 \end{smallmatrix} \right) $ and $0=\left( \begin{smallmatrix} 0\ 0\\ 0\ 0 \end{smallmatrix} \right)$. Thus $B_2 = \{a, b, ab, ba, 0\}$. Furthermore, the relations $aa=bb=0$, $aba=a$ and $bab=b$ suffice to recover completely the multiplication in $B_2$. If one adds the identity matrix, one gets the 6-element monoid $B^1_2$. The semigroups $B_2$ and $B_2^1$ are nicknamed the universal counterexamples because they provide many counterexamples in semigroup theory.