There is something I have been wondering about recently when thinking about a SciFi scenario: What would happen if Apostolic Succession was broken - some catastrophic event killed, among many other people, all bishops in the Catholic Church (and perhaps even all priests and deacons). Would there be any way for the Church to continue?
Well, let's get the boring answer out of the way; I think that an event that ends the line of Apostolic Succession would most probably be an event that ends the Cosmos itself. :P
But bracketing that for now, I imagine one way around this would be to go the route that the Church of Jesus Christ of Latter-Day Saints did: have an event that somehow "rekindles" Apostolic Succession. The church claims that Joseph Smith was ordained by the Apostles Peter, John, and James sometime around 1830, and definitely no later than 1835, in order to revive the priesthood, which had hithertofor gone extinct. (Adherents of many Sufi orders make similar claims about how their spiritual lineages started with Muhammad visiting their founders in dreams and initiating them into mystical brotherhoods that the founder then transmitted to others).
Now, of course, the question becomes: would people be convinced by such a claim? (I'll leave the truth value of the claim aside). I think that people who don't pay much attention to the hierarchy in the first place may accept it without question, if things remain more or less in continuity on the ground level. But this may be an opportunity for Protestants and (assuming their bishops weren't also wiped out) Orthodox to make some major inroads on what remains of the Catholic lay faithful, since I really do think this would really destroy the credibility of Catholic truth claims.
Instead of the "rekindling" approach, it's also possible that surviving lay faithful may take up positions similar to Sedevacantists today, placing heavier emphasis on para-liturgical celebrations and devotions; the Rosary could become the central communal act of worship, for example. For those who want sacraments (and again, this is assuming other lineages of Apostolic Succession survive), perhaps members of this remnant community may join communities of Orthodox Christians, either truly assimilating or becoming crypto-Catholics.
Or maybe they take a position like the one argued by James Carroll in his article, Abolish the Priesthood, where he basically argues that Catholics should reframe the Mass so that "Christ is experienced not through the officiant but through the faith of the whole community." I would say that Carroll is an apostate and a heretic who is advocating for the Church to commit suicide — but this might be an attractive option for people living in a world where Apostolic Succession has died, but who nonetheless want to maintain a Catholic identity.
There's actually a real sect like this in Orthodoxy - the bezpopovtsy (priestless) Old Believers.
For background, in the 17th century, Patriarch Nikon of the Russian Orthodox Church made several liturgical reforms to bring Russian Orthodoxy in line with Greek practice, most famously changing how the fingers should be arranged in the sign of the Cross. This caused several people to be upset and split off, becoming the Old Believers, consisting of two factions. The popovtsy (priested) believed that the mainline Russian Orthodox Church was in grave error but still basically legitimate, and so maintained their own clergy consisting of unemployed or banished clergy - think of them as a Russian version of the SSPX.
Meanwhile, the bezpopovtsy taught that the Nikonite reforms signalled that the Great Apostasy was here and the Antichrist (who they identified as a spiritual force rather than a person) now ruled the world, and so the priesthood had been taken from earth, making them the Russian version of sedevacantists. Since Orthodox doctrine allows laity to perform baptism and prescribe penance if no priest is available, they did just that with their lay leaders. As you might expect, their communities were very small, very isolationist and very prone to splitting, but both them and popovtsy Old Believers are still around.
