What is the best symbol for vector/matrix transpose?

It's always difficult to answer questions for "the best" or "most popular". As is mentioned, these are typically opinions. But you did say that your objection was the fact that the "T" symbol was too big. Therefore, I would recommend the \intercal symbol to produce a "T" which isn't so big. Also, writing the vectors and matrices in bold seems, in my opinion, to make it look a little better. Try the following code:

\documentclass{article}
\usepackage{amsmath,amsfonts,amssymb}
\begin{document}
$\mathbf{A}^\intercal$\\
$\mathbf{c}^\intercal \mathbf{x}$\\
$c^T x$\\
$\mathbf{M}^\top$
\end{document}

The Comprehensive LaTeX Symbol List says the following:

Some people use a superscripted \intercal for matrix transpose: A^\intercal. (See the May 2009 comp.text.tex thread, "raising math symbols", for suggestions about altering the height of the superscript.) \top, T, and \mathsf{T} are other popular choices.

In order to give some reference:

(DIN) EN ISO 80000-2:2013 writes it like the following.

% arara: lualatex

\documentclass{article}
\usepackage{mathtools}
\usepackage{unicode-math}
\setmathfont{XITS Math}
\newcommand*{\matr}[1]{\mathbfit{#1}}
\newcommand*{\tran}{^{\mkern-1.5mu\mathsf{T}}}
\newcommand*{\conj}[1]{\overline{#1}}
\newcommand*{\hermconj}{^{\mathsf{H}}}

\begin{document}
\[\matr{A}\tran\]   

\[\matr{A}\hermconj\coloneqq(\conj{\matr{A}})\tran\]
\end{document}

The good part here is that the 'big' T visually fits to the H of the same size which might be used for the Hermitian conjugate matrix.