The most noteworthy complication, and most heard about in the news, is anaphylaxis.
This article breaks down cases of anaphylaxis nicely: Reports of Anaphylaxis After Receipt of mRNA COVID-19 Vaccines in the US—December 14, 2020-January 18, 2021. They provide a great table with a specification of whether the case happened during the 15-minute window:
So for ...
The data on this aren't hard to find. Here are some more examples that specifically provide time frames to back up @A Rogue Ant's answer:
For the Pfizer vaccine:
During December 14–23, 2020, monitoring by the Vaccine Adverse Event
Reporting System detected 21 cases of anaphylaxis after administration
of a reported 1,893,360 first doses of the Pfizer-...
For the UK up to 27 October 2021:
All recipients were requested to wait 15 minutes before departing the point of vaccination.
A history of anaphylactic reactions to any of the ingredients forbids taking the vaccine, but which dose (first or second) the listed reactions occurred after is not specified in the data.
Estimated first/second doses - Vaccine ...