Many times my computer will not disconnect from the internet when I sign out of MSN. There is no icon in the tray indicating the connection or lack thereof.
How can I check to see if my modem is  connected?
Is it possible that someone is using my computer?
If so, they must have left an instruction on my machine.

   I doubt anyone is hacking into your computer and using it.  That would not keep you from disconnecting.
   Are you using dialup, cable, or dns?

I am using dial-up, XP home, hp 7935, MSN 9. MSN takes a long time to develope when I sign in from this computer. I have a win 98 computer which is much faster.

   There should be a setting somewhere that specifies whether MSN will disconnect the dialup connection when you log out or not.
   You can always go to the Control Panel section of Windows Explorer (right click on the Start button in Windows and select Explore from the popup menu), expand the Control Panel folder, and look in the Network Connections folder.  That should show your dialup connection and show whether you are connected or not.  You should be able to disconnect from there.
   I say, should, because I no longer have dialup, but that is where I remember the dialup connections to be.

I can disconnect from network connections, but it puzzles me why it doesn't disconnect when I sign out of MSN, why it doesn't disconnect when I click this on the sign-in screen and why I don't see an icon in my tray.

   I don't know.