Ensure that you have a known-good game cartridge inserted before powering on the SNES console. Most older, cartridge-based game consoles don't boot without a game installed.
Does the power light to the left of the controller ports turn on when you turn on the power switch?
If not, check your power adapter and outlet for functionality and good connection.
Check that the reset switch is not stuck. Wiggle it back and forth through its range of motion a few times. If something sticky gets in there it can get jammed in the reset position, which will prevent any game from booting.
You mentioned that there is no image. Are you getting any sound from your games after waiting a few minutes with everything connected and turned on?
If you hear game sounds or music from the SNES, but see no picture and you've already (really and truly) ruled out the cabling and TV, then you might have a bad main board in the console.
If you don't get any video or sound, then it's not booting your game. You can find other recommendations about how to clean game and console connections, but usually inserting and ejecting a game 10-20 times will knock loose or grind off enough gunk to get a game to boot. Try this, and be sure to use the eject lever instead of ripping the game out to avoid console connector damage. If it's still not working, repeat with another game.