Proof-of-burn handles node selection by selecting among nodes that can prove they have burned some coins (i.e. sent them to a verifiably un-spendable address) some time ago. Selected nodes receive the transaction fees in the block. The ‘some time ago’ condition prevents a miner waiting until they see a block with lots of transaction fees, and then mining that block. The recent act of burning could be undone by a blockchain re-organization, and the miner could then be able to burn the same coins again to grab another block. The burn should be far enough ago that there is no practical possibility that it could be undone.
Burning coin is expensive from a node’s point of view, more expensive than just holding a stake. Therefore, it provides a disincentive for a node to cheat by signing blocks in parallel chains (i.e. it deals with the ‘nothing-at-stake’ problem). How much burning will happen? Miners will burn cryptocurrencies at an average rate very close to the average rate that ordinary users are sending them fees, minus the miners’ true real-resources costs (i.e. hardware, electricity and the like for handling transactions and blocks and burn proofs). These costs will be lower than proof-of-work related costs.
Strengths - Consumes no resources other than the burned underlying asset. Weaknesses - Only works where there is a currency that has already been mined.
Under Proof-of-capacity, the node selected to create the next block is chosen through a pseudorandom process that depends partly proving that it has allocated a non-trivial amount of memory or disk space to solve a challenge. So, it shares some similarities to Proof-of-work; instead of CPU functions being allocated it is memory hard functions; each HDD megabyte is an additional lottery ticket in the mining.
Strengths - As there is less variation in memory access times than in CPU speeds, Proof-of-capacity is ‘fairer’ than Proof-of-work, and runs less risk from mining centralisation. Miners have monetary incentive to store useful public data in the network. Weaknesses - There is still the problem of ‘nothing-at-stake’; a node can momentarily scan its terabytes to try its chances on an alternative chain. So, many chains can be mined simultaneously without spending resources.