Do you mean you see the blue OSMC splash screen ?
What happens if you press ALT-F2 to try to switch to another virtual console, do you get a login prompt ?
Here are two possible methods to get to an emergency shell where you should be able to capture logs.
Put your SD card in a PC and edit the file cmdline.txt, at the end of the line add a space then:
recovery
save the file then try to boot, you should be taken to an emergency console. If that does not work, try again but instead of recovery add:
systemd.unit=emergency.target
Assuming you boot to a command prompt, run the following commands to gather logs and save them to the fat partition:
/usr/bin/grab-logs -A -C
sync
Then shut down, put the SD card back in your PC and you should find a file called uploadlog.txt - open this in a text editor and paste it into paste.osmc.io and provide the link here.