The two most common issues with USB drives are power consumption and dodgy USB controllers/bridges in the USB drive.
First make sure the drive is powered properly (RPi can only supply a very limited amount of power). If you are using a drive with it's own power supply then you should be fine. If you are using a bus-powered drive and connect it to a powered USB hub.
If your drive has some dodgy USB controller then first try with the current LE11 nightly build. It comes with a lot newer linux kernel and might already have workarounds for the dodgy controller in place.
If this doesn't help and/or you want to run LE10 then first try booting from a SD card (or your USB key), plug in the drive and post a full log (run "pastekodi" via ssh or use the log upload function) - this will give us some information about your drive.
You can also check dmesg/journal or lsusb output yourself.
The info you/we need are the manufacturer and model of the drive and the USB vendor and product id.
Searching for that eg on the RPi forum https://forums.raspberrypi.com/ might already give you the info you need - usually some usb quirks that you need to add to cmdline.txt.
The most common issue is broken UAS support in the drive, in that case you need to add something like usb_storage.quirks=1234:5678:u to /flash/cmdline.txt (where 1234:5678 need to be changed to the USB vendor/product ID of your drive).
so long,
Hias