On a RedHat/CentOS system, I can verify the installed RPMs on my system using rpm -V.
What is the Ubuntu or apt equivalent of this command?
The package "debsums" is what you want to install to perform hash checks against installed packages.
For example, to check for changed files:
sudo debsums -c
Note that not all packages ship with md5sum file lists. You can see a list of those on your system with:
sudo debsums -l
To followup to what Kees said, to deal with packages that don't include a buildtime generated debsums md5sum file, by default the debsums package also installs an apt hook to generate a debsums as part of the package installation process. Thus, one way to generate the debsums file for those packages that are missing them is to apt-get install --reinstall them.