DeclareMathOperator and widearcarrow with kpfonts

Replace \newcommand instead of \DeclareMathOperator and your code works perfectly.

For the differences between \newcommandand \DeclareMathOperator I suggest to see this link: newcommand vs. DeclareMathOperator

\documentclass{article}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{kpfonts}
\newcommand{\dSing}{\widearcarrow{\mathrm{Sing}}}
\newcommand{\sSing}{\widehat{\mathrm{Sing}}}
\begin{document}    
\[\dSing(X) \quad \widearcarrow{Sing}\quad\sSing(A)\]
\end{document}

Apparently, math accents from family 3 don't work in \operatorname.

You can use a lower level interface:

\documentclass{article}

\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{kpfonts}

\DeclareRobustCommand{\dSing}{%
  \mathop{\widearcarrow{\mathrm{Sing}}}\nolimits
}
\DeclareMathOperator{\sSing}{\widehat{Sing}}

\begin{document}

\[
\dSing(X) \quad \widearcarrow{Sing} \quad \sSing(A)
\]

\end{document}

\renewcommand{\d}[1]{\widearcarrow{\text{#1}}}
\def\dSing{\d{Sing}}

Don't know why it works this way, but it does. May be a feature of the definition of widearcarrow in the package.