Standalone Clients
For total stand alone Synchro SVN is a powerful and cross platform solution. It looks like the most native application on each of the platforms.
The Subversion website includes a listing of other standalone SVN Clients (most are cross platform). [Copied list below from http://subversion.tigris.org/links.html#clients]
eSvn - cross-platform QT-based GUI frontend to Subversion
http://sourceforge.net/projects/esvn
FSVS - fast subversion command-line client centered around software deployment
http://fsvs.tigris.org/
KDESvn - A Subversion client for KDE
http://www.alwins-world.de/wiki/programs/kdesvn
QSvn - A cross-platform GUI Subversion client
http://ar.oszine.de/projects/qsvn/
RapidSVN - A cross-platform GUI front-end for Subversion
http://rapidsvn.tigris.org/
RSVN - Python script which allows multiple repository-side operations in a single, atomic transaction.
https://opensvn.csie.org/traccgi/rsvn/trac.cgi/wiki
SmartSVN - A cross-platform GUI client for Subversion
(Not open source. Available in a free and a commercial version.)
http://www.syntevo.com/smartsvn/
Subcommander - A cross-platform Subversion GUI client including a visual text merge tool.
http://subcommander.tigris.org/
SvnX - A Mac OS X Panther GUI client.
http://www.lachoseinteractive.net/en/community/subversion/svnx/
Syncro SVN Client - Cross-platform graphical Subversion client.
(Not open source. Free trial versions available for Mac OS X, Windows and Linux.)
http://www.syncrosvnclient.com
WorkBench - Cross platform software development GUI built on Subversion written in Python
http://pysvn.tigris.org/
Versions - A GUI Subversion client for Mac OS X.
(Not open source; requires commercial license.)
http://www.versionsapp.com/
ZigVersion - a Subversion Interface for Mac OS X. Aims to design an interface around the typical workflows of programmers.
(Note that this is not open source.)
http://zigversion.com/
Integrated Clients
TortoiseSVN is the best general use system [An integrated system is not standalone - Thanks Martin Kenny]. It integrates itself into Windows Explorer (You can use it in explorer or any shell dialog) so it works extremely well and gives you the full power of SVN.
Ankhsvn is a good solution that integrates into Visual Studios (Except Express Editions).
SVN Notifier monitors your repositories and will notify you when anything changes. It integrates with TortoiseSVN to show you diffs and commit logs. Very handy when working in a team environment.