How to allow non-superusers to mount any filesystem?

There are a couple approaches, some of them mostly secure, others not at all.

The insecure way

Let any use run mount, e.g., through sudo. You might as well give them root; it's the same thing. The user could mount a filesystem with a suid root copy of bash—running that instantly gives root (likely without any logging, beyond the fact that mount was run).

Alternatively, a user could mount his own filesystem on top of /etc, containing his/her own copy of /etc/shadow or /etc/sudoers, then obtain root with either su or sudo. Or possibly bind-mount (mount --bind) over one of those two files. Or a new file into /etc/sudoers.d.

Similar attacks could be pulled off over /etc/pam.d and many other places.

Remember that filesystems need not even be on a device, -o loop will mount a file which is owned (and thus modifiable) by the user.

The mostly secure way: udisks or similar

The various desktop environments have actually already built solutions to this, to allow users to mount removable media. They work by mounting in a subdirectory of /media only and by turning off set-user/group-id support via kernel options. Options here include udisks, udisks2, pmount, usbmount,

If you must, you could write your own script to do something similar, and invoke it through sudo—but you have to be really careful writing this script to not leave root exploits. If you don't want your users to have to remember sudo, you can do something like this in a script:

#!/bin/bash
if [ $UID -ne 0 ]; then       # or `id -u`
    exec sudo -- "$0" "$@"
fi

# rest of script goes here 

The will-be-secure someday way: user namespaces

Linux namespaces are a very lightweight form of virtualization (containers, to be more specific). In particular, with user namespaces, any user on the system can create their own environment in which they are root. This would allow them to mount filesystems, except that has been explicitly blocked except for a few virtual filesystems. Eventually, FUSE filesystems will probably be allowed, but the most recent patches I could find don't cover block devices, only things like sshfs.

Further, many distro kernels have (for security reasons) defaulted to not allowing unprivileged users to use user namespaces; for example Debian has a kernel.unprivileged_userns_clone that defaults to 0. Other distros have similar settings, though often with slightly different names.

The best documentation I know of about user namespaces is an LWN article Namespaces in operation, part 5: User namespaces.

For now, I'd go with udisks2.

You can do it, but you need to modify the entry in /etc/fstab corresponding to the filesystem you want to mount, adding the flag user to this entry. Non-privilege users would then be able to mount it.

See man mount for more details.

Here is the wiki for configuring polkit rules for udisks/udisks2 in order to mount partitions by non-root (e.g. users) group.

Save the code below to /etc/polkit-1/rules.d/50-udisks.rules

polkit.addRule(function(action, subject) {
  var YES = polkit.Result.YES;
  var permission = {
    // only required for udisks1:
    "org.freedesktop.udisks.filesystem-mount": YES,
    "org.freedesktop.udisks.filesystem-mount-system-internal": YES,
    "org.freedesktop.udisks.luks-unlock": YES,
    "org.freedesktop.udisks.drive-eject": YES,
    "org.freedesktop.udisks.drive-detach": YES,
    // only required for udisks2:
    "org.freedesktop.udisks2.filesystem-mount": YES,
    "org.freedesktop.udisks2.filesystem-mount-system": YES,
    "org.freedesktop.udisks2.encrypted-unlock": YES,
    "org.freedesktop.udisks2.eject-media": YES,
    "org.freedesktop.udisks2.power-off-drive": YES,
    // required for udisks2 if using udiskie from another seat (e.g. systemd):
    "org.freedesktop.udisks2.filesystem-mount-other-seat": YES,
    "org.freedesktop.udisks2.encrypted-unlock-other-seat": YES,
    "org.freedesktop.udisks2.eject-media-other-seat": YES,
    "org.freedesktop.udisks2.power-off-drive-other-seat": YES
  };
  if (subject.isInGroup("users")) {
    return permission[action.id];
  }
});

Assume you are in the "users" group, using the following command to mount a partition (no need sudo).

# udisks2
udisksctl mount --block-device /dev/sda1

# udisks
udisks --mount /dev/sda1