Strange. I have hibernate and suspend worked. Just tested. My LMDE laptop have:
$ uname -a
Linux mycomp 2.6.39-2-686-pae #1 SMP Tue Jul 5 03:48:49 UTC 2011 i686 GNU/Linux
$ dpkg -l | grep linux-image-686
ii linux-image-686-pae 2.6.39+35.1 Linux for modern PCs (meta-package)
$ cat /etc/apt/sources.list
deb
http://packages.linuxmint.com/ debian main upstream import backport
deb-src
http://packages.linuxmint.com/ debian main upstream import backport #Added by software-properties
deb
http://debian.linuxmint.com/latest testing main contrib non-free
deb
http://security.debian.org/ testing/updates main contrib non-free
deb
http://www.debian-multimedia.org testing main non-free
deb-src
http://debian.linuxmint.com/latest testing main contrib non-free
deb-src
http://security.debian.org/ testing/updates main contrib non-free
deb-src
http://www.debian-multimedia.org testing main non-free
"Update Manager" -> "Update Pack Info" -> "Latest Update Pack used by this system: 2"
Swap present.
$ cat /etc/fstab
...
UUID=438e97e6-f175-4a1b-8b2f-f4af986077ff none swap sw 0 0
...
$ cat /etc/initramfs-tools/conf.d/resume
RESUME=UUID=438e97e6-f175-4a1b-8b2f-f4af986077ff
$ free
...
Swap: 8193020 0 8193020
$ sudo blkid
...
/dev/sda6: UUID="438e97e6-f175-4a1b-8b2f-f4af986077ff" TYPE="swap"
I use XFCE and gdm3. Have no separate record in grub to map to swap.
Maybe this information could help.