How large is the space of almost convergent sequences?

We consider the subspaces $c$, $\widehat{c}$, $S$ of ${\mathrm{\ell }}^{\mathrm{\infty }}$, where $\widehat{c}$ consists of almost convergent sequences, and $S$ consists of sequences whose arithmetic means of consecutive terms are convergent. We know that $c\subset \widehat{c}\subset S$. We examine the largeness of $c$ in $\widehat{c}$, $\widehat{c}$ in $S$ and $S$ in ${\mathrm{\ell }}^{\mathrm{\infty }}$. We will do it from the viewpoints of porosity, algebrability and measure.