Which solo artist has had the most number one hits, and who has had the second most number one hits?

Mariah Carey.

Carey's latest single, "Touch My Body," made music history this week as she surpassed Elvis Presley as the solo artist with the most No. 1 singles in the modern pop era.

The cut is her 18th No. 1 song and the first single from her "E=MC2" album, due April 15. Carey bests a tally Presley has held for 40 years. He last topped the singles chart with "Suspicious Minds" in 1969. Carey's 2005 comeback, "The Emancipation of Mimi," spawned two No. 1 hits, "We Belong Together" and "Don't Forget About Us." Should the new album meet with the same degree of success, she could tie or even best the Beatles' 20 No. 1 singles, a record for any pop group.

Justin Timberlake and Fergie share the record for the most number-one singles in Pop 100 for a solo artist, having four chart-toppers.
[en.wikipedia.org/wiki/List_of_Pop_100_chart_achievements_and_trivia#Artist_with_the_most_.231_singles_.28solo_artist.29]
[starswelove.com/scriptsphp/news.php?newsid=7022] links no longer exist

http://articles.latimes.com/2008/apr/03/entertainment/et-albums3

If you're intested in the UK charts, then Elvis has had a record 21 British chart-toppers, interestingly with 18 different songs (some were reissued/remixed). The Beatles come second with 17 No 1s.

Paul McCartney has featured on more #1s than any other artist albeit under many different guises and pseudonyms: 24 in total!
- solo (1) "Pipes Of Peace"
- Wings (1) "Mull Of Kintyre/Girls' School"
- with Stevie Wonder (1) "Ebony And Ivory"
- with The Christians, Holly Johnson, Gerry Marsden, Stock Aitken & Waterman (1) "Ferry 'Cross The Mersey"
- as part of Ferry Aid (1) "Let It Be"
- as part of Band Aid (1) "Do They Know It's Christmas?"
- as part of Band Aid 20 (1) "Do They Know It's Christmas?"
- The Beatles (17).
[theofficialcharts.com/stats-most-no1.php] link no longer exists

http://www.everyhit.com/record1.html

George Strait beats Mariah, Justin and Fergie by a COUNTRY MILE (yes, pun intended)!

George has garnered 57 No. 1 songs on all country charts, and has sold more than 73 million recordings. His certifications from the RIAA include 13 multi-platinum, 32 platinum and 35 gold albums. His overall certification numbers are third of all musicians, behind Elvis Presley and The Beatles.

ref: Kingsley, Paul (1995). George Strait. Liner Notes for Strait Out of the Box: MCA Records.
ref: Stark, Phyllis (February 26, 2006). "Being George Strait". American Chronicle.

It's now 60 number one hits and 100 million in sales:
https://theboot.com/george-strait-no-1-songs