Pyrosequencing was developed in the mid 1990's and by the year 2000 massive parallel sequencing using this technique was becoming quite common. This "NextGen" sequencing technique was behind the massive explosion in sequences in the early part of the 21st century.2
Even newer techniques are available today and there's a debate about whether they should be called Third Generation Sequencing (Heather and Chain, 2015).
Nature has published a nice review of the history of DNA sequencing (Shendure et al., 2017). I recommend it to anyone who's interested in the subject. The figure above is taken from that article.
1. Many labs were using the technology in 1976 before the papers were published.
2. New software and enhanced computer power played an important, and underappreciated, role.
Heather, J.M., and Chain, B. (2015) The sequence of sequencers: The history of sequencing DNA. Genomics, 107:1-8. [doi: 10.1016/j.ygeno.2015.11.003]
Maxam, A.M., and Gilbert, W. (1980) Sequencing end-labeled DNA with base-specific chemical cleavages. Methods in enzymology, 65:499-560. [doi: 10.1016/S0076-6879(80)65059-9]
Sanger, F., Nicklen, S., and Coulson, A.R. (1977) DNA sequencing with chain-terminating inhibitors. Proceedings of the National Academy of Sciences, 74:5463-5467. [PDF]
Shendure, J., Balasubramanian, S., Church, G.M., Gilbert, W., Rogers, J., Schloss, J.A., and Waterston, R.H. (2017) DNA sequencing at 40: past, present and future. Nature, 550:345-353. [doi: 10.1038/nature24286]