PeterC and Jack Cleveland, you both ask very interesting questions.

The answer is a little complicated. 
Much radio material survives - some from as far back as the early 30s, and a great deal from the late 40s.
All of the regular radio shows (Philco, Chesterfield, GE and CBS) from the commencement of the Philco radio shows in 1948 also survive in good quality audio. 

Much of this has been released in some form either officially or unofficially, with the Crosby family responsible for covering nearly all the CBS material over the last few years, together with individual song tracks from earlier years.
Many of the complete radio shows have circulated among collectors but have usually only been released as single song tracks (though some LPs were unofficially issued of complete shows in the '70s and '80s), but you will realise that much has not. The range is vast.

There is quite a lot of material, literally "in the vaults" some of which has been released by the Crosby family in recent years but the precise extent is not known, and the suggestion has been made that some might yet have to be identified. The range is potentially wide - private recordings - alternative takes, discarded items etc. Issues of "regular" material from this source have frequently included such "lost" items as bonus tracks. 

You will learn a lot by browsing through the discography here, which lists many known items and the issues on CD (and a few LPs).
Some very early material has been issued on CD by ICC and I believe remains available. 

However, with the issue of the CD mentioned by Archiefit "On the Sentimental Side" there is no known substantial body of commercial studio recordings that has not been issued in some form.  
There are some known individual tracks that have not been issued officially. 

And then there are quite a few items in the form of fluffs that have seeped out into the collecting community. 

And that is before we get onto the subject of video.

 

Last edited by Richard Baker (16/1/2019 10:18 am)

The last stage performance was on Monday October 10, 1977, at the Conference Centre in Brighton, Sussex, England.

The last recording date for radio (songs later released on LP "The Final Chapter") 11 October 1977.
BBC Studios, Maida Vale, London, England.

If you wish to distinguish between recordings made for radio and those made solely for record release then there were studio sessions on 14th September for "Seasons" (at which some readings were also recorded) and 4th October at which two tracks were recorded for dubbing into an LP otherwise made at a live performance in Norway "Joe Bushkin Celebrates 100 Years of Recorded Sound". Again take a look at the Discography.
http://www.bingmagazine.co.uk/bingmagazine/crosby.html

I believe Frank Sinatra recorded over 1500 songs, but the figure could be up to 1800, Bing around 2300. Methodology for either could be challenged depending on how such items as medleys and alternate takes are counted.

These figures do not include radio material where Bing is streets ahead. 

My Ella discography is laid out in a way not easily conducive to arriving at any total but I don't believe her figures are quite as high, though in her case such statistics are difficult to get at because many of her live performances and radio performances were recorded with an eye on potential record release. Most were never released but are included in the discography I have as if they were studio recordings.

I would comment though that unheard of or totally forgotten singers of the early era of recording leave these figures in the shade. Those singers spent all their time singing anonymously or under varying pseudonyms and had phenomenal output, with Henry Burr sometimes mentioned as having made as many as 12,000 records. 
Many of the dance band singers of the pre 1940 era spent their time accompanying numerous bands (many of which used varying identities), racking up very large figures.

PeterC has noted that Bing's recording statistics are are stupendous. As an Aussie, I'll throw in another statistic. In the first part of the 20th century, the Australian baritone Peter Dawson devoted much of his career to performing and recording in the United Kingdom. He is estimated to have recorded close to 2000 titles, many using more than two dozen pseudonyms. He recorded onto Edison cylinders, 78s and long-playing records.

I had never heard of Peter Dawson before.  I am sure this is the Peter Dawson you refer to:

https://en.wikipedia.org/wiki/Peter_Dawson_(bass-baritone)

I am sure there are a few other artists who have recorded more than Bing, Frank, Ella, and the others.  But I doubt any who have done so are of their stature.  But wow!  This Peter Dawson had quite the career.

Since my contribution on 22nd, I've looked again at the notes which accompany a Decca 5-CD set issued a couple of years ago titled FROM MELBA TO SUTHERLAND. AUSTRALIAN SINGERS ON RECORD which includes seven tracks by Peter Dawson. The notes state that it is estimated that he made some 3500 recordings and to have sold some 14 million records between 1904 and 1954, of material ranging from opera and oratorio to popular ballads and music hall songs. (Yes, Ron, a greater variety than Bing.)