I'm trying to install Solus on my PC, probably going to be my first distro that isn't Ubuntu based. Now, I've had a similar problem installing Apricity OS.

So after I've burned the ISO onto a USB, I booted into my USB. I chose the option to proceed into Solus to try it, but it just went into a blank screen and the monitor eventually turned off. The same thing actually happened with Apricity. I noticed that this didn't actually happen only when I used an Ubuntu based distro. My friend suggested that I disable secure boot or look for an option to use legacy boot.

After looking in my motherboard manual, I found an option to use legacy boot, but still doesn't work, same issue occurs. I haven't found an option to doable secure boot however and I'm not sure where the real problem is actually at.


Motherboard: MSI X99a SLI Plus

This appears to happen often with PCs having Nvidia graphics, the nouveau driver appears a little unstable lately.
BUT:
As for what issues your live distribution(s) are facing is hard to pinpoint without some kind of information regarding what hardware your "PC" is comprised of. Motherboard is a start, but listing graphics adapter is likely the best hardware component to list when having graphic issues.
Seeing as it appears to boot and give you an option to try it means it's not bios settings or motherboard.

Well, my GPU is an Nvidia GTX 970. I'm not sure if that's where the problem is, but I'm not able to get into the OS in the first place. I can't really try it, it just doesn't take me into it.

As I suspected, I have the 970M
To boot into graphical desktop, you should be able to edit the kernel parameters at the grub menu. For Debian, since I did not use live media, I used the install media after installation in rescue mode to edit the /etc/default/grub file and added --> nouveau.modeset=0 as a parameter, ran command update-grub as root after editing it and rebooted to a desktop with less than optimal resolution, but had a desktop environment. Editing that file this way would get it to boot properly every time after, eventually I installed Nvidia.run installer from Nvidia's web site after installing kernel headers to get the nice desktop. I prefer from repositories but the 340 drivers I need (for desktop with GTS 8600) from Jessie are not compatible withe the version of xorg in my Stretch installation, which is why I went with Nvidia.run installer.

So...
For a live system, you should try adding either of the three parameters to the kernel line before booting.
The first one works with my Debian Stretch, if it don't do it for you, try the second one. I have not booted this way since I made the change permanent, but theoretically it should be the same only it would be a one time only, and would have to be done every time you boot it.

Ok, so I tried the second option (modeset nouveau.modeset=0) which ended up working so I can try out Solus. Haven't tried the third option for the parameters yet, but I might. Now I noticed that the resolution is stuck at 720p, but my monitor is a 1080p monitor. I tried to change the resolution, but I can't seem it. I assumed that I needed to install my GPU drivers... but I have no idea where to find that.