In terms of promising technologies, they're broadly categorisable as thermal, kinetic, battery/fuel cell, and thermochemical. Most of the promising ones are far enough along the learning curve that other markets (such as green hydrogen/ammonia for fertiliser driving electrolysers and small scale/more efficient chemical reactors) will drive the learning curve.
Thermal storage concepts include:
Molten salt thermal. short/medium for high grade heat. Most high grade heat is dispatchable (fire) and so doesn't make sense to store, or expensive (solar thermal, nuclear) and so isn't worth pursuing.
Sand thermal batteries. Low grade heat for medium/long term. Only useful for heating and some industrial purposes. Has a minimum size (neighborhood). Literally dirt cheap.
Thermochemical. I guess this is kind of a fuel? Use case is for low grade heat so it can go here. Phase change materials like sodium acetate or reversible solution like NaOH seem really appealing for heating. Back of envelope says it's close to competitive with electric heating, so I'd expect more attention as it's cheaper than any technology that stores work. No idea why it isn't being rolled out. You could even charge it with heat pumps for extremely high efficiency if needed.
Kinetic:
Lifting stuff. Only really works for water without large subsidies and only if you already have at least one handy reservoir like a watershed or cavern. No reason to expect it would suddenly get cheaper as digging holes and moving big things is already something lots of industries try to do cheaply. Great addition to existing hydro.
Sinking stuff (using buoys to store energy). I can't comprehend how this can be viable. I have seen it espoused, but it doesn't pass back of the envelope test unless I did a dumb.
Squashing stuff. Compressed air energy storage. Tanks are just barely competitive with last gen batteries capacity-wise, efficiency isn't great. There are concepts for underwater bladders (let the watter do the holding) or cavern based storage that seem viable at current rates. Achievable with abundant materials so worst case scenario we nut up and spend$500/kWh. Key word CAES, cavern or underwater energy storage
Battery/fuel cell:
Lithium ion: One of the best options currently. Will be heavily subsidised by car buyers. Has hit limits of current mining production which puts a floor on price and is ecologically devistating.
X ion where x is probably sodium: Great slot in replacement. Barring large surprises will expect it to replace LiFePO4 very soon for most uses. Expect the learning rate of lithium ion manufacturing to continue resulting in a sharp jump to $60/kWh in 2021 dollars and eventual batteries around $30/kWh. Key word natron (have just brought their first product to market and are working with other parts of the supply chain to scale up)
Flow batteries, air batteries and fuel cells. These are almost the same concept. You have a chemical reaction that makes electricity with a circular resource like hydrogen, methane, ammonia, or electrolyte. Downside is most versions require a prohibitive amount of some metal like rutheneum or vanadium or something. Not a fundamental limit, but not sure it will be a great avenue as research goes back a fair ways. Aluminum-air batteries are one interesting concept. Essentially turning Al smelters into fuel production facilities. Keywords iron-air aluminum-air, redox-flow, direct methane fuel cell, ammonia fuel cell, ammonia cracking, nickel fuel cell.
Molten salt batteries. Incredibly simple, cheap and scalable concept that has no problems with dendrites (and so theoretically no cycle limit) with one limitation on portability (they must be hot, sloshing is bad) and one as yet insurmountable deal breaking flaw (incredibly corrosive material next to an airtight insulating seal). Look up Ambri for details of an attempt which has presumably failed by now. There is a more recent attempt using a much lower temperature salt and sodium sulfur which shows promise. Keywords ambri, sodium sulfur battery.
Thermochemical:
Any variation on burning stuff you didn't dig up.
Hydrogen is hard to store more than a few days worth, but underground caverns could help. I expect a massive scandal about fugitive hydrogen, toxicity and greenhouse effect in the 2030s sometime. It's borderline competitive to make now. Main limitation is cost of energy (solved by more wind and solar and more 4 hour storage) and cost of capital (platinum/palladium/rutheneum/nickel are usually required). Lots of work going on to reduce the latter and to increase power density and efficiency. If you were directing a billion dollars of public funds this would probably be the place to put it. Keywords $200/kw electrolyser, hysata 95% efficient.
Methane, ammonia, dimethyl ether, methanol, etc. These are all far easier to store than hydrogen. Production needs large scale but is borderline viable already if you have cheap hydrogen. Keywords ammonia energy storage, synthetic fuels, efuels, green ammonia, direct ammonia electrolysis.
Then there's virtual batteries.
Many loads like aluminum smelting can be much more variable than they are now. Rearranging workflows such that they can scale up or down by 50% and change worker tasks to suit has the same function as storage during any period where consumption isn't zero. EV's can kinda fit here too and kinda fit actual storage (especially if they power other things)
Biofuels. Not technically storage, more dispatchable, but it serves a similarfunction. Bagasse is an option for a few percent of power. Waste stream methane is a possibility for a couple % of power. Limited by the extremely low efficiency of photosynthesis so something PV based will likely be a better way of making hydrocarbons from air and sunlight. Most other 'biofuels' are either fossil fuels with extra steps or ways of getting paid green energy credits for burning native forests. Some grad student might surprise us by creating a super-algae that's 10% efficient and doesn't all get eaten if there's a single bacterium in the room. Detangling it all is hard, but I wouldn't be surprised if wind + solar + biofuels + reigning in the waste was enough -- it certainly works for some people doing off grid.
I'd expect a system based on sodium ion (or even lithium) batteries and synthetic fuels to render any fossil fuel mix unviable in the next decade or two. More scalable batteries or scalable fuel cells would hasten this somewhat.