The top 125 Peel sessions have been selected by the people who knew John best - his family, friends, studio engineers, producers and musicians. (BBC Keeping It Peel site.) 
When asked what his favourite sessions were by Serbian station B92, Peel was unwilling to commit himself to more than a couple of vague preferences:
My favourites would be fairly obscure things - the two sessions the Slits did during the punk era which were just magical, I thought, were just terrific. Oh, there have been so many. There have been so few that have been bad, it’s amazing, really, when you consider how many have been done. Many thousands now. Very few of them have been disappointing. The Clash did half one, and then amazingly said that the equipment in the studio wasn't up to the standards that they'd expected so they couldn't complete the session. Which seemed to me to be unbearably pretentious of them (laughs). It'd be very difficult to pick out an absolute favourite from them. There was one by the reggae band Culture that out of all of the sessions that were released on record is the one that I listen to the most, I think. (MTB Interview).
In the event, it was left to others unnamed to make the final choice, presented by the BBC as an alphabetical list, thereby eschewing the need for an indication, given the above 'criteria', of picking a top 10. These are presented below in the same format: dates are those of the original recording. Note that each artist has been allotted only one entry.
The list is as notable for what it omits as for those it includes. The Culture session praised by JP above is curiously absent, as is the most frequently repeated session of all, the first by Siouxsie & The Banshees.
↑Rather unhelpfully, this site does not specify who was involved in selecting this list. However, since those involved in its construction appear to have had more than a passing relationship with Peel, it was felt that its inclusion here was warranted.
↑Strictly speaking, not eligible for inclusion on this list as it was a live recording from the Queen Elizabeth Hall. Listed by the BBC as a session, but not by Ken Garner in The Peel Sessions.