Looks to me like the camera is being forced to use high ISO due to incorrect settings being used for the conditions it's in.
Op should first select a low light option (like night portrait) so the camera can use a wide aperture (F3.5) and slower shutter to keep ISO (noise) under control, failing that the flash should be used.
It seems point and shoots are not a magic bullet, sometimes some photographic knowledge and skill is required also.
I assumed the OP knew they aren't very good at high ISO so didn't bother checking

A Samsung WB500 or Sony HX5 will offer better low light performance at £100 and £170 respectively, for example:
TZ10 @ 1600ISO - http://images.trustedreviews.com/images/article/inline/12830-tz10iso1600.jpg
HX5 @ 1600ISO - http://images.trustedreviews.com/images/article/inline/13003-hx5iso1600.jpg
I am taking the old sony and the lumix. Luckily the sony can fire while plugged in, another thing the lumix can't do.
It's more about 'feature' lists than quality...how did we manage before face detection...?