これらの壊れた依存関係を修正するにはどうすればよいですか？

aptitudeは依存関係が壊れたパッケージがあることを教えてくれます：

kmiller@debian-dev-box:~$ sudo aptitude search '.*' | grep ^.B
iB  console-setup-linux             - Linux specific part of console-setup      
iB  gir1.2-freedesktop              - Introspection data for some FreeDesktop co
iB  gir1.2-glib-2.0                 - Introspection data for GLib, GObject, Gio 
iB  gir1.2-gstreamer-0.10           - Description: GObject introspection data fo
iB  gir1.2-json-1.0                 - GLib JSON manipulation library (introspect
iBA libatk-adaptor                  - AT-SPI 2 toolkit bridge                   
iB  libgcc1                         - GCC support library                       
iBA libgfortran3                    - Runtime library for GNU Fortran applicatio
iBA libgomp1                        - GCC OpenMP (GOMP) support library         
iB  libstdc++6                      - GNU Standard C++ Library v3               

しかし、次の理由により、「壊れた」という状態が悪くなることはありません。

kmiller@debian-dev-box:~$ sudo apt-get -f install
Reading package lists... Done
Building dependency tree       
Reading state information... Done
0 upgraded, 0 newly installed, 0 to remove and 0 not upgraded.

kmiller@debian-dev-box:~$ sudo apt-get -u dist-upgrade
Reading package lists... Done
Building dependency tree       
Reading state information... Done
Calculating upgrade... Done
0 upgraded, 0 newly installed, 0 to remove and 0 not upgraded.

libgcc1の依存関係が壊れているのが最も心配です。

何を修正すべきかを見つけるにはどうすればよいですか？

kmiller@debian-dev-box:~$ sudo aptitude install -f
...
0 packages upgraded, 11 newly installed, 177 to remove and 0 not upgraded.
...
The following packages have unmet dependencies:
 libstdc++6 : Breaks: gcc-4.3 (< 4.3.6-1) but 4.3.5-4 is to be installed.
 gir1.2-json-1.0 : Conflicts: gir1.0-json-glib-1.0 but 0.10.2-2squeeze1 is to be installed.
 gir1.2-gstreamer-0.10 : Conflicts: gir1.0-gstreamer-0.10 but 0.10.30-1 is to be installed.
 libgfortran3 : Breaks: gcc-4.3 (< 4.3.6-1) but 4.3.5-4 is to be installed.
 libgomp1 : Breaks: gcc-4.3 (< 4.3.6-1) but 4.3.5-4 is to be installed.
 libgcc1 : Breaks: gcc-4.3 (< 4.3.6-1) but 4.3.5-4 is to be installed.
 gir1.2-freedesktop : Conflicts: gir1.0-freedesktop but 0.6.14-1+b1 is to be installed.
 gir1.2-glib-2.0 : Conflicts: gir1.0-glib-2.0 but 0.6.14-1+b1 is to be installed.
 libatk-adaptor : Conflicts: at-spi but 1.32.0-2 is to be installed.
 console-setup-linux : Breaks: console-terminus but 4.30-2 is to be installed.

apt-getそして、aptitude異なる依存リゾルバを持っています。壊れたパッケージをで修正するための提案をaptitudeに提供できますaptitude install -f。更新された質問から判断すると、にリリースやディストリビューションが混在しているようですsources.list。