SD card won't stay in the slot of my Panasonic Lumix DMC-SL80

The DMC-FZ20 works with MMC and SD cards, 2GB or smaller. Your 8GB card is an SDHC card, which fits into the slot but is electronically incompatible.

I have the exact same problem, have not found a solution yet.

No. According to the camera specifications, the DMC-FZ20 works with SD cards. While SDHC cards will fit into the slot, they will only work if the device has been designed to work with them.
You'll need to use SD cards, 2GB and smaller. You also don't need to worry about speed. Any SD card you can buy now will be fast enough to keep up with the camera.

There are no SD memory cards 4GB or larger because the Secure Digital standard limits card capacities to 2GB. While SDHC cards will fit into the slot, they are electronically incompatible and will not work in the DMC-FZ30. You will need to stay with SD cards 2GB and smaller.

The DMC-LC33 is designed to work with SD cards. Your 4GB card is actually an SDHC (Secure Digital High Capacity) card. While it fits into the slot it will not work. You need to use SD cards, 2GB and smaller.

Remove the SD card and slide the lock switch on it to the unlocked position. If it's already there then slide it fully to locked and then fully back again.

If this fails then you need to try another SD card to see whether it's a camera fault. If the camera is at fault then there's nothing more you can do other than to return the camera to Panasonic for repair.

Does the PC pop up with anything when the camera is plugged in? if so, make sure the camera is in plyback mode and check windows explorer, under My computer, there should be a extra drive letter pop up.
If not then the USB socket can become damaged easily (have a look). If the socket is damaged, then you can either get it repaired or buy a SD card reader. that stays connected to the PC through it's own USB cable, you take the SD card out of the camera and plug it in to the reader (this is a better and cheaper option)
Cheers.

You need to format the card in the camera to optimize it.

See fix 4 http://camerarepair.blogspot.com/2009/07/simple-fix-for-memory-card-locked-error.html?m=1