Edited to include those commented:
Here are a list of Gobblegums that I know to be available through digging with a golden shovel. I haven't heard of any others being found yet and have received several of these multiple times. Can anybody confirm any aside from these?

Classic:
-Always Done Swiftly
-Alchemical Antithesis
-Sword Flay
-Coagulant

Mega:
-Licensed Contractor
-Board to Death
-Slaughter Slide

Rare:
-Bullet Boost
-Crate Power

Ultra:
-Head Drama
-Profit Sharing