So I installed linux mint (17.3) the other day and now i can't seem to get my headphones or even microphone working. I only managed to get sound out of my monitor when it was connected with an hdmi cable.
OK. Try this: cat /proc/asound/cards It should show all your audio connectors. Do you see your headphone jack? If this output is empty, ALSA cannot find your headphone jack and therefore no headphone connected to it.
How old is your PC? - Linux Mint uses a relatively old kernel (= 3.13 - current stable version is 4.5.3) and builds on top of Ubuntu 14.04 (O.o WTF?!) - which maybe has no driver support for your connector.
You can upgrade the kernel from the updates gui (view > linux kernels), no need to jump on another distro. If you installed the proprietery graphics drivers, make sure to uninstall them before you upgrade the kernel, then reinstall later
Since I have been distro-hopping, sound doesn't work in most distros out of the box for me. I have 3 sound cards in addition to 2 inbuilt. What I end up doing is launch some long (1 hour) youtube-video while troubleshooting.