\documentclass{article}
\begin{document}
$$ a=b \]
but
\[ a=b $$
  

\[
  a=b $$ and $$ b=c
\]

\begin{equation}
  a=b \] and \[ b=c
\end{equation}

\begin{equation}
  a=b $$ and $$ b=c
\end{equation}

\begin{equation}
  a=b \] and $$ b=c
\end{equation}

\begin{equation}
  a=b $$ and \[ b=c
\end{equation}

\begin{equation}
  a=b
  \footnote{But beware that
    \[ b=c \]
    and so on.}
\end{equation}

\begin{equation}
  a=b
  \footnote{But beware that
    $$ b=c $$
    and so on.}
\end{equation}

\end{document}

% These are not legit, however
\begin{equation}
  a=b $ and $ b=c
\end{equation}

\begin{equation}
  a=b \) and \( b=c
\end{equation}

\begin{eqnarray}
  a&=&b\\ \] and \[
  b&=&c\\
\end{eqnarray}

\end{document}