With molten fuel, a Molten Salt Reactor would generate 4,000 times less mining waste and up to 10,000 times less nuclear waste than any solid fueled reactor. The fission byproducts can be easily extracted, so the fuel can fission completely.

Solid-fueled Light Water Reactors, to make 1 gigawatt-year electricity need 250 tons (250,000 kg) uranium (incl 1.75 tons U-235), make 35t enriched uranium (1.15t U-235). This would leave 215 tons depleted uranium (0.6t U-235), 35 tons spent fuel (33.4t U-238, 0.3t U-235, 1.0t fission products, 0.3t plutonium). Kirk Sorensen TEAC3

Molten Salt Reactors (e.g. LFTR) to make 1 gigawatt-year electricity: need 600 to 800 kg (0.8 ton) of Thorium or any isotope of Uranium. D. LeBlanc / Nuclear Engineering and Design 240 (2010)

The uranium (or plutonium or other transuranic elements) are completely fissioned in a MSR. (Thermal or epithermal spectrum MSR can fission U233, U235, Pu239; fast spectrum MSR can fission all isotopes of uranium and transuranic elements. All MSR can breed Th to U233 or U238 to Pu239.)

83% of the waste (fission byproducts) from a LFTR are safely stabilized within 10 years. The remaining 17% (135kg for a GigaWatt-year) are elements that need to be stored less than 350 years to become completely benign. 135kg vs 250 tons (250,000kg) from a LWR.

A 40-megawatt test reactor running for 10 years would “burn” 141 Kg. U-233, and produce less than 1 milligram of plutonium or other transuranic elements. Leave these inside the reactor, where neutron bombardment will cause them to fission. Charles Holden, TEAC 2011

The traditional “nuclear waste problem” includes storing all the uranium with the fission byproducts. However, separating the uranium (and plutonium) is easy. We don’t separate them because of the “reprocessing is bad” conversations. Storing uranium in low density so chain reaction is impossible is relatively simple and produces low levels of radiation. Reprocessing by PUREX is very complex, and produces fuel that is enriched enough for making new LWR fuel pellets; the enrichment of depleted fuel is harder than starting from raw uranium. Separating the uranium, plutonium and other fissionable elements from the fission products, and storing them separately, doesn’t require any enrichment.

Many fission byproducts are very short-lived (half lives in seconds to months), and therefore highly radioactive. As the separated fission products have much smaller volume, they can be left as salts and allowed to solidify and decay in short-term storage. (Other storage can work, such as vitrified in glass, which is used at some storage sites like Hanford.)

“FS-MSRs can be employed to consume actinides from light-water reactor (LWR) fuel or, alternatively, to extend fissile resource availability through uranium-to-plutonium breeding. FS-MSR reactors are highly flexible and can be configured into modified open or full-recycle configuration. The modified open FS-MSR fuel cycle options do not include chemical processing of the fuel salt… The conversion ratio of an FS-MSR is largely determined by the fissile-to-fertile-material ratio in its fuel salt. Thus, a single reactor core design may be capable of performing both fissile resource extension and waste disposition missions.”
Fast Spectrum Molten Salt Reactor Options, Oak Ridge National Laboratory, July 2011