Info about distro: 

DISTRIB_ID=Ubuntu
DISTRIB_RELEASE=14.04
DISTRIB_CODENAME=trusty
DISTRIB_DESCRIPTION="Ubuntu 14.04.1 LTS"
NAME="Ubuntu"
VERSION="14.04.1 LTS, Trusty Tahr"
ID=ubuntu
ID_LIKE=debian
PRETTY_NAME="Ubuntu 14.04.1 LTS"
VERSION_ID="14.04"
HOME_URL="http://www.ubuntu.com/"
SUPPORT_URL="http://help.ubuntu.com/"
BUG_REPORT_URL="http://bugs.launchpad.net/ubuntu/"

using Xubuntu 4.10


Info about greeter: 

Package: lightdm-gtk-greeter
Status: install ok installed
Priority: optional
Section: x11
Installed-Size: 914
Maintainer: Xubuntu Developers <email address hidden>
Architecture: amd64
Version: 1.8.5-1ubuntu2~ubuntu14.04.1
Replaces: lightdm (<< 0.9.2-1), lightdm-gtk
Provides: lightdm-greeter
Depends: libc6 (>= 2.4), libcairo2 (>= 1.2.4), libgdk-pixbuf2.0-0 (>= 2.22.0), libglib2.0-0 (>= 2.30.0), libgtk-3-0 (>= 3.1.4), libido3-0.1-0 (>= 13.10.0daily13.06.19), libindicator3-7 (>= 0.4.90), liblightdm-gobject-1-0 (>= 1.4.0), libx11-6
Recommends: desktop-base, gnome-themes-standard, gnome-icon-theme-symbolic, policykit-1
Breaks: lightdm (<< 0.9.2-1), lightdm-gtk
Conffiles:
 /etc/lightdm/lightdm-gtk-greeter-ubuntu.conf a5f919e30b751b6840339d6366b46e83
Description: simple display manager (GTK+ greeter)
 A LightDM greeter that uses the GTK+ toolkit.
Homepage: https://launchpad.net/lightdm-gtk-greeter
Original-Maintainer: Debian Xfce Maintainers <email address hidden>

with a lightdm 

Package: lightdm
Status: install ok installed
Priority: optional
Section: x11
Installed-Size: 544
Maintainer: Robert Ancell <email address hidden>
Architecture: amd64
Version: 1.10.1-0ubuntu1
Provides: x-display-manager
Depends: debconf (>= 0.5) | debconf-2.0, sysv-rc (>= 2.88dsf-24) | file-rc (>= 0.8.16), libc6 (>= 2.14), libgcrypt11 (>= 1.4.5), libglib2.0-0 (>= 2.37.3), libpam0g (>= 0.99.7.1), libxcb1, libxdmcp6, libpam-runtime (>= 0.76-14), libpam-modules, adduser, libglib2.0-bin, dbus, plymouth (>= 0.8.8-0ubuntu6.1)
Pre-Depends: dpkg (>= 1.15.7.2)
Recommends: xserver-xorg, unity-greeter | lightdm-greeter | lightdm-kde-greeter
Conflicts: liblightdm-gobject-0-0, liblightdm-qt-0-0
Conffiles:
 /etc/init.d/lightdm 506283f07afc03bc4d74437fac2372dd
 /etc/dbus-1/system.d/org.freedesktop.DisplayManager.conf b76b6b45d7f7ff533c51d7fc02be32f4
 /etc/pam.d/lightdm-greeter 175dc5fc5903124a38f7a8d74b9a3def
 /etc/pam.d/lightdm-autologin e192ee66bdfbf54fc7d869379f02f8ff
 /etc/pam.d/lightdm fcc2191b2fc7fab2b0d723faec5635cd
 /etc/lightdm/users.conf 9365d0d580a33cc9be3b5ad77ce5dcc8
 /etc/init/lightdm.conf 0288f8c2077484afe55a74b81c4dea31
 /etc/apparmor.d/lightdm-guest-session 7d122d506d65910f9252b5b36aeb13a1
 /etc/apparmor.d/abstractions/lightdm 673415141c85dd8e03a24edd30124b02
 /etc/apparmor.d/abstractions/lightdm_chromium-browser 54532af6815c8096323201cdb1946544
Description: Display Manager
 LightDM is a X display manager that:
  * Has a lightweight codebase
  * Is standards compliant (PAM, ConsoleKit, etc)
  * Has a well defined interface between the server and user interface
  * Cross-desktop (greeters can be written in any toolkit)
Homepage: https://launchpad.net/lightdm

Malfunction started indeed after an update, but I tried to reinstall all packages related to lightdm and lightdm-gtk-greeter without success.