Most people who ask this question have one of three things in hand: an old deed from a parent or grandparent, a letter from a landman or a mineral buyer, or a royalty check that arrived out of nowhere. Whichever it is, the path in Utah is the same four steps.
Most of Utah is described from the Salt Lake Meridian and Base Line (township north or south, range east or west of Salt Lake City); the heart of the Uinta Basin uses the Uintah Special Meridian instead, so a Duchesne or Uintah County tract can read T3S R4W with either meridian — the deed has to say which. The description is on the deed, on any division order you signed, and usually printed in the property column of a check stub. Write down every tract, including any government lots (irregular parcels along rivers and section edges, described as "Lot 3" rather than a quarter-quarter).
Mineral deeds, reservations and leases are recorded with the County Recorder of the county the land sits in — not with the state. Utah's 29 county recorders keep the grantor/grantee index; Uintah and Duchesne — the Uinta Basin, where most of the state's wells are — and San Juan, Grand and Carbon carry most of the mineral instruments, and the larger offices publish their indexes online. You are looking for the instrument that gave your family the minerals (a mineral deed, or a warranty deed that reserved the minerals when the surface was sold) and for anything that later conveyed them away. If the trail runs through a death, look for a probate, a decree of distribution or an affidavit of heirship in the same index; minerals that were never formally passed on are the most common reason a check stops or never starts.
Utah is a public-lands state: roughly two-thirds of the wells sit on federal (BLM Vernal, Moab and Price field offices) or Ute Tribe trust minerals, and the School and Institutional Trust Lands Administration (SITLA) holds state trust minerals in the checkerboard sections. Private fee minerals are the minority, concentrated in parts of the Uinta Basin and along the Wasatch Front. If you have a Utah check, the first question is whether your interest sits under a federal, tribal, SITLA or fee lease — the deductions and the paperwork differ for each. The map shades federal minerals so you can see at a glance whether the section you own surface on has private minerals at all.
Once you have a tract, look at it. Utah Division of Oil, Gas and Mining (Department of Natural Resources) records will tell you which wells have been permitted, drilled and produced on that section and, through spacing orders, which horizontal wells a mile away are draining it. If there is production and you are not being paid, the operator's division-order department is the next call; if there is a rig on the way, a landman will find you before long. Utah's activity is concentrated in the Uinta Basin (Uintah and Duchesne counties — Natural Buttes gas and the Altamont-Bluebell and Monument Butte oil fields, now drilled horizontally in the Uteland Butte and Wasatch) and the Paradox Basin in the southeast (Greater Aneth in San Juan County, Cane Creek in Grand); if your tract is there, assume someone else already knows about it.
Put the tracts, the recorded instruments and the fractions in one place. Owners who can answer "what do I own and where" in one sentence get paid faster, negotiate better leases and hand their heirs something other than a shoebox. If you own in more than one state — common for families who homesteaded across the Rockies — keep one set of books per person or trust and tag each tract with its state.
No. Ownership is proven by documents recorded with each county recorder. DOGM records show wells and production, not who owns the minerals under them. Federal mineral ownership is a BLM record; tribal minerals are a BIA record.
Probably. Operators pay the owner of record, often found through a title search after a well is drilled. Ask the payor for the division order and the legal description it is based on, then confirm the chain of title at the county.
It depends on whether they are producing, what the wells are making, how much undrilled acreage sits inside an active spacing unit, and current prices. Look at the section's production history and nearby spuds before you answer a buyer's letter.