Third Update
Using \dimeval{…} instead of \dimexpr … \relax and \Ifstr instead of \ifstr, as suggested by @cabohah in a comment below.
\documentclass{scrreprt}
\RedeclareSectionCommands[
tocbreakafternumber,
tocindent=0pt
]{chapter,section,subsection,subsubsection}
\renewcommand\chapterlinesformat[3]{\Ifstr{#2}{}{}{#2\\*}#3}
\renewcommand*{\sectionlinesformat}[4]{%
\hspace*{#2}%
\parbox{\dimeval{\linewidth-#2}}{\raggedsection
\Ifstr{#3}{}{}{#3\\}%
#4}%
}
\usepackage[english]{babel}
\usepackage{blindtext}
\begin{document}
\tableofcontents
\listoffigures
\Blinddocument
\begin{figure}%
Figure
\caption{A Figure}%
\end{figure}
\end{document}
Second Update
Since version 3.20 you can use option tocbreakafternumber in \RedeclareSectionCommands to get a break after the sectioning numbers TOC.
\documentclass{scrreprt}[2016/04/09]
\RedeclareSectionCommands[
tocbreakafternumber,% <-
tocindent=0pt
]{chapter,section,subsection,subsubsection}
\renewcommand\chapterlinesformat[3]{\ifstr{#2}{}{}{#2\\*}#3}
\renewcommand*{\sectionlinesformat}[4]{%
\hspace*{#2}%
\parbox{\dimexpr\linewidth-#2\relax}{\raggedsection
\ifstr{#3}{}{}{#3\\}%
#4}%
}
\usepackage[english]{babel}
\usepackage{blindtext}
\begin{document}
\tableofcontents
\listoffigures
\Blinddocument
\begin{figure}%
Figure
\caption{A Figure}%
\end{figure}
\end{document}
The result is the same as in the pictures below.
Update
Since version 3.19 the recommended way to change the order or the layout of the sectioning numbers and text is redefining commands \chapterlinesformat, \chapterlineswithprefixformat, \sectionlinesformat and \sectioncatchphraseformat.
Additional I have tested the hack/workaround in my original answer for the TOC with the pre-release of the next KOMA-Script version (3.20). With this upcoming version simple redefining \numberline has no effect. You could appended code to \numberline but at the moment I would suggest to redefine \addtocentrydefault. Maybe in the future there will be a special tocstyle that defines your needed style for the toc entries automatically.
\documentclass[toc=flat]{scrreprt}[2015/10/03]
\renewcommand\chapterlinesformat[3]{\ifstr{#2}{}{}{#2\\*}#3}
\renewcommand*{\sectionlinesformat}[4]{%
\hspace*{#2}%
\parbox{\dimexpr\linewidth-#2\relax}{\raggedsection
\ifstr{#3}{}{}{#3\\}%
#4}%
}
\renewcommand\addtocentrydefault[3]{%
\ifstr{#2}{}{%
\addcontentsline{toc}{#1}{\protect\nonumberline#3}%
}{%
\addcontentsline{toc}{#1}{\protect\nonumberline#2\\*#3}%
}%
}
\usepackage{blindtext}
\begin{document}
\tableofcontents
\listoffigures
\Blinddocument
\begin{figure}%
Figure
\caption{A Figure}%
\end{figure}
\end{document}


Original answer:
Somebody has asked Markus Kohm a similar question. So there is now a suggestion for the headings on the KOMA-Script website (German):
\documentclass[chapterprefix]{scrreprt}
\renewcommand*{\chapterformat}{\thechapter\autodot}
\RedeclareSectionCommand[innerskip=0pt]{chapter}
\makeatletter
\renewcommand*{\raggedsection}{\raggedright\renewcommand*{\@hangfrom}[1]{##1\\*}}
\makeatother
\usepackage{mwe}
\begin{document}
\Blinddocument
\end{document}
Note: This code is a workaround and uses KOMA-Script internals. It works with the current version 3.18 But maybe it will not work with future releases.
Additionally here is a workaround/hack for the table of contents also based on an idea of Markus:
\documentclass[chapterprefix,
toc=flat
]{scrreprt}
\renewcommand*{\chapterformat}{\thechapter\autodot}
\RedeclareSectionCommand[innerskip=0pt]{chapter}
\makeatletter
\renewcommand*{\raggedsection}{\raggedright\renewcommand*{\@hangfrom}[1]
{##1\\*}}
\makeatother
\BeforeStartingTOC[toc]{\renewcommand*{\numberline}[1]{#1\\*}}
\RedeclareSectionCommands[tocnumwidth=0pt]{chapter,section,subsection}
\usepackage{blindtext}
\begin{document}
\tableofcontents
\listoffigures
\Blinddocument
\begin{figure}%
Figure
\caption{A Figure}%
\end{figure}
\end{document}

Or without toc=flat:

titlesecis pretty straightforward.