Tonight, or this morning, a number of elections are particularly close. From my own guess, I think the margins in Pennsylvania (20 electoral votes) and New Hampshire (4 electoral votes) look to be the closest. In the event that these elections prove to be decisive in a presidential candidate's quest to secure 270 electoral votes, recounts are likely. (Of course, if a candidate concedes, or if the margins become unrealistically large, or if there are still more tipping point states, or there are others, this prediction is sure to be wrong!)
Many recount procedures begin administratively through the Secretary of State or an election board. It's worth noting with any contested election that usual basis for filing a claim is in state court. For all the attention given to federal causes of action, the ordinary act of recounting is typically reserved to state courts. In Florida in 2000, the federal issues were raised as a counter to the procedures implemented by state courts; they were not originally federal causes of action. It's likely, then, that we would expect basic recount procedures to arise in state court; federal claims could arise in that context. It's less likely that claims would originally be filed in federal court, but there are plausible scenarios where it might happen (and where federal courts would not abstain from exercising jurisdiction, another issue!), but I'll focus on the state-based claims at the moment.
It's also worth noting that after Bush v. Gore, the key deadline this year is December 13. That's the date set by federal law for presuming the regularity of the election results in Congress. Most opinions in Bush v. Gore recognized that this deadline was an important date for a state to meet (although some disputed how important or what procedures should take place in the late days before that deadline). Any litigation, then, would occur with an eye toward that date.
Below are the general timing and triggers for basic recounts, but there are obviously many more conditions and opportunities that can arise beyond these. But as we hardly know if there will be challenges, much less here, I'll stick with a couple of bare outlines of order and timing.
An automatic recount is triggered under Pennsylvania law if the margin of victory is one-half a percentage point or less. (25 Pa. Stat. § 3154(g)(1)(ii)) (There is no recount if the losing candidate requests no recount.) The Secretary issues an order by 5 pm ET November 17. (25 Pa. Stat. § 3254(g)(2)) The recount and recanvass must be finished by noon ET November 29. (25 Pa. Stat. § 3254(g)(5)) (Some more background is here.)
There are no "automatic" recounts in New Hampshire, but parties can petition for a recount if the margin is less than 20%. (N.H. Rev. Stat. § 660:1) (Some more background is here.) A candidate must file a petition by November 11. The process to recount must begin by November 16. (N.H. Rev. Stat. § 660:4)