Yes those designs are already in use today. Modern reactors are incredibly safe. The only modern disaster was Fukushima and that didn't even cause any deaths and was brought on by a tsunami.
And here is your source. And this is for ALL nuclear waste in the world.
The volume of high-level radioactive waste (HLW) produced by the civil nuclear industry is small. The IAEA estimates that 392,000 tonnes of heavy metal (tHM) in the form of used fuel have been discharged since the first nuclear power plants commenced operation. Of this, the agency estimates that 127,000 tHM have been reprocessed. The IAEA estimates that the disposal volumeb of the current solid HLW inventory is approximately 29,000 m3.1 For context, this is a volume roughly equivalent to a three metre tall building covering an area the size of a soccer pitch.