Maximal Abelian von Neumann algebra Let $H$ be the Hilbert space of complex-valued functions on $\mathbb{R}$ that are square integrable w.r.t. the Lebesgue measure.
What is a concrete maximal von Neumann algebra of $\mathcal{B}(\mathcal{H})$?

There are lots of masas in $B(H)$. They can be classified in two kinds, discrete and continuous (and and direct sums thereof).
The canonical example of a continuous masa in your setting would be $L^\infty(\mathbb R)$, seeing as multiplication operators.
The canonical example of a discrete masa is the diagonal masa: you fix an orthonormal basis $\{e_n\}$, and consider the corresponding orthogonal projections $\{E_n\}$. Then
$$
A=\{\sum_ka_kE_k:\ a\in\ell^\infty(\mathbb N)\}
$$
would be the diagonal masa corresponding to the orthonormal basis $\{e_n\}$. Not that you gain anything, but if you want to  make this concrete, you can take $\{e_n\}$ to be the Hermite Polynomials. Or you can use a double index and define
$$
e_{n,m}=e^{2\pi in(x-m)}\,1_{[m,m+1)},\qquad n,m\in\mathbb Z.
$$
This would make
$$
(E_{n,m}f)(x)=\langle f,e_{n,m}\rangle\,e_{n,m}=\bigg(\int_m^{m+1}f(t)\,e^{-2\pi i (t-n)}\,dt\bigg)\,e^{2\pi in(x-m)}\,1_{[m,m+1)}.
$$
In this case $A$ would consist of the operators
$$
(T_af)(x)=\sum_{n\in\mathbb Z}a_{n,m}\,\bigg(\int_m^{m+1}f(t)\,e^{-2\pi i (t-n)}\,dt\bigg)\,e^{2\pi in(x-m)},\qquad x\in[m,m+1),
$$
where $a\in\ell^\infty(\mathbb Z^2)$.
