Safari won't open in Yosemite
I would check Activity Monitor while you are clicking the Safari icon. See if there are any processes like "Safari" or "Safari Web Content" that is showing up, and see if there is a red "not responding". This probably will not happen if the app does not even bounce, but it's worth a try. You can also try basic things like right clicking, clicking "Show all windows" or going to Options and clicking asking to this desktop. Also, have you been working with multiple displays recently? The safari window may be "stuck" on the other display, so you should check it out by plugging any back in.