How do you determine if 580B backhoe hydrostatic drive or shuttle?

Hi Frank:
Hydrostatic drive?
Shifter 3 position - Forward Neutral Reverse - Lever?
Follow the linkage from the shift lever. Probably a worn part or broken spring.

Sounds like your teeth on the ring gear / fly wheel are broken / chipped or damaged. Inspecting the ring gear = Take a long flat tip screw driver and place the tip in between the cogs at the highest point possible, having the driver at 9 & 3 O'Clock postions. Now pry up until you are at the 10 or 11 O'Clock positions, The flywheel will only rotate an inch or two. Make sure you are in neutral. This may take a few minutes but it's the only way you can inspect by your self.

if the engine rpm dropped with the reduction in speed, I would be checking for brakes being applied
as for the hard shifting , that could be a shift problem in the transmission
call the cat technician to have the fault codes read first to determine the fault area

The ring gear is bolted to what is called a ring gear carrier, the carrier is bolted inside the rear end housing with two bolts on each side, it is shimmed on each side to adjust backlash and pinion depth so DON'T mix up the left and right carrier shims, once the carrier is out with the ring gear attached you can remove the pinion shaft and drive out the bearing races of the pinion bearings. You also need to remove the pinion flange where the drive shaft U-Joint attaches, just hammer it off with a brass faced hammer (a steel hammer will run the flange) once the large center nut is removed.

1971 case 580ck manual shuttle is locked up . when I let out the clutch it stalls the motor

Starter motor spins, but does not engage

1. Check the starter motor for a seized or binding pinion gear.
2. Remove the flywheel inspection plate and check for a damaged ring gear.

check the connection of your control circuit

I have a 1974 case backhoe 580 it will not move in any gear.

Your Onan engine will do excellent with Mobil One 0w40, Mobile One 5w30.

DO NOT use hydraulic oil in the transmission/transaxle, you will destroy it. Your tractor uses an Eaton 11 Hydrostatic transmission pump with the oil resevour being the cast iron transaxle. The WheelHorse transaxle is as rugged as they come. It uses a solid cast iron housing, case hardened intermediate gears and forged differential gears with an eight pinion differential.

Your transaxle requires 10w40 synthetic or conventional oil as well as a wix 51410 filter or the OEM wheelHorse filter. It should be changed every 100 hours with conventional oil or 250 hours with synthetic. DO NOT use hydraulic oil!

Is it a hydrostatic drive and did you pull the pin that is located by the hitch. The pin should be pushed in in order to engage hydrostatic drive.