Can I use SPARSE somehow on a non-nullable bit column with mostly false values?
A filtered index (WHERE IsMostRecentRun = 1) sounds like a better idea to me than using sparse. If you can make it so that false is instead represented by null, you may be able to do both, but while that will potentially save some space in the base table, I suspect the bigger gain would be in query performance from the filtered index - as long as it's covering. If you need too many columns to cover and/or need to also filter or join on other columns, then you may find a balancing act between the improvements you get from the seek or range scan on the filtered index and the costs of things like lookups to get at the rest of the columns.

That all said, it seems suspicious you would need an additional column to serve as a flag for this. Isn't this information redundant? Can't it be determined by other data? In which case I would focus on index tuning to optimize the use of the existing columns instead of storing and maintaining redundant data.

(Also I don't think Pinal is correct.)