Well, to start with, Treasury securities will likely go poof, and with those trillions, banks worldwide would go under. Social Security is significantly paid for through federal bonds; that would drop the bottom out of the existing pension net. The Federal Reserve will lose all except 9% of its value, because it's backed by American treasury notes: that is the primary way through which the dollar would die. Furthermore, when the Federal Reserve collapses, the Federal Reserve notes also lose all value. All sorts of money funds break apart, and this isn't a matter just for rich people; anyone who's planned for retirement through a Roth IRA or similar plans is going to get hammered almost as badly as Social Security people. China possibly tanks as a third of the People's Bank's reserves threatens to vanish into the aether, which could tip its construction bubble and any other less-obvious instabilities in the economy over the edge; Japan's banks dream of the days of stagflation as it realizes a fifth of its value is similarly at risk (by the bye, this is part of the reason why these two nations were among the most vocal the last time this came around in 2013). European banks are not exactly stable yet, either, and will not be pleased to have the bottoms drop out from under them again. Anything that runs on the dollar is going to go through a hiccup; dislocation in the energy market through the break of the petrodollar will likely hurt Iran and Russia less than Venezuela or Nigeria, but this is likely more of a short-term issue unless multiple potential reserve currencies attempt to secure dominance (say, the Euro versus the Ruble). The US, as a major market, will stop importing as much as domestic production for export becomes more competitive with a weakening dollar (no storm cloud without a silver lining...) and also as domestic consumption in an unstable market tanks under a sort of "mattress-stuffing mentality" (...which is sometimes incoming lightning), hurting any economies that export significant amounts to the US - China, Mexico, Canada, and the combined EU most prominent among them. The biggest issue is that interest rates will rise globally because one of the fundamental underpinnings of the modern global investment system, the "risk-free" nature of U.S. Treasury bonds, will vanish. Everyone who holds those, which is to say, everyone will try to unload them somewhere, and there aren't going to be that many buyers. As interest rates rise, economic recovery will vanish as investment slows.
It does, however, depend significantly on the scale of the default. For example, if the US writes off its foreign debts without touching its domestically-held debts, it doesn't solve everything, but it does still cause vast amounts of harm abroad which will rebound domestically in terms of not only loss of diplomatic credibility, but economic credibility - there's no reason why, with the precedent set, a second default couldn't wipe out the rest of those promissory notes. It's something that can be recovered from, as we can see from the EU's slow recovery, but as you say, it will be no picnic.