I don’t see carbon extraction from the atmosphere being a leading option unless it could somehow be made much cheaper than the alternatives, and I haven’t seen anyone propose even a theory as to how that could happen. I do think chances are good we will do carbon extraction from seawater, but mostly as a fuel stock (also fertilizers, but that’s not really a form of sequestration). Where biological sequestration is concerned, there is some potential in modifying agricultural practices, reforestation, and doing land management to combat, and possibly reverse, desertification. (On the latter count, I really hope Allan Savory is onto something, but my gut tells me it won’t be anywhere near that simple.) And maybe some nutrient seeding in currently-barren parts of the oceans to promote diatom blooms and other sorts of shell formation.
Where mineral sequestration is concerned, the potential is large, but not with our current system. Natural rock weathering removes about a billion tons of CO2 from the air each year, and we know of several ways to mimic that process or help it along, but the problem is that pretty much every strategy involves a major energy input–such as grinding and dispersing large amounts of olivine, or doing high-pressure injection of large volumes of seawater through peridotite formations. If we were to find ourselves with a source of abundant, cheap, clean energy, that would improve the viability of a number of these strategies greatly. But of key importance is to develop the clean energy first, and to make the transition off of fossil fuels.
I see no good sequestration strategies which could make a significant dent in our overall greenhouse footprint if we do not get off the fossil fuels first, and I see no good strategies which have unbounded potential at the level which makes them good strategies. Many of the best strategies look like they would have diminishing effectiveness, such that the easiest sequestration occurs at first, and then it gets progressively more difficult. For such strategies, it would be best to treat them as finite resources, and using up their potential while we are still emitting large amounts of fossil carbon would only diminish their ultimate long-term potential for the sake of allowing us to continue to burn more fossil carbon for a while longer. (And it would be doubly pointless if we were to actually expend dirty energy to drive those sequestration processes.)
So, even if there could, in theory, be some reasons for optimism about the long-term potential of sequestration, it seems to me that at this point there are even more compelling reasons not to evince such optimism.