Schnorr sigs will be a big deal. Work is set to start this year, but as OmegaStarscream states, that could mean we're looking at sometime around 2020 till Schnorr sigs are activated on the network (it can apparently be achieved by a soft fork).
It might be an exaggeration to say Schnorr solves the scaling problem, but it will certainly help alot (supposedly 25-30% extra space will be made available on-chain*). Privacy will improve for certain transactions too.
* proponents of on-chain scaling will, of course, be against Schnorr sigs for some invented non-reasons
Bitcoin's already facing too much challenges making it lose its market dominance. Last time bitcoin hit all-time-high it was practically unusable and people moved to litecoin and ethereum to carry out transactions.
I wouldn't worry too much, Lightning will make a very large difference to the transaction capacity long before that. It's important to remember that Bitcoin's market position is mostly driven by real demand, not so easily said for most other cryptocurrencies. Also important to remember that Bitcoin is the leader because of it's the most highly used cryptocurrency network, and it has the engineering quality needed to support that level of use.