General information about apprenticeships can be found on the National Archives website: Looking for records of an apprentice or master.
This index on FindMyPast was derived from the Stamp Duty registers, held by the National Archives in series IR 1. The images from which the index was derived can be viewed for free on the National Archives website, by browsing the Discovery catalogue to the relevant year and piece. Note that the same records have been indexed on Ancestry.co.uk since 2011, and Ancestry.co.uk includes both an index and images.
I doubt there is a definitive list of occupations.
In a research paper titled Apprenticeship and Occupations in Southern England, 1710-1760, using the Stamp Duty registers (IR 1) as their source, the author analyzes the number of each occupation for three counties (Surrey, Sussex, and Wiltshire). The data can be viewed on pages 16-17. The list from this paper includes all occupations from these three counties that occurred in these records 5 or more times. I have put the list in alphabetical order:
Also, on the Guildhall Library website is one list of companies in London, and includes most of the major occupations that would be apprenticed:
Apothecaries, Armourers and Brasiers
Bakers, Barbers (ca. 1540-1745 Barber Surgeons), Basketmakers,
Blacksmiths, Bowyers, Brewers, Broderers, Butchers
Carmen, Carpenters, Clockmakers, Coach and Coach Harness Makers,
Combmakers, Cooks, Coopers, Cordwainers, Curriers, Cutlers
Fanmakers, Farriers, Fellowship Porters, Feltmakers, Fishmongers,
Fletchers, Founders, Framework Knitters, Fruiterers
Gardeners, Girdlers, Glass Sellers, Glaziers, Glovers, Gold and Silver
Wyre Drawers, Grocers, Gunmakers
Joiners and Ceilers
Masons, Merchant Taylors#, Musicians
Painter Stainers, Parish Clerks, Pattenmakers, Paviors, Pewterers,
Pinmakers, Plaisterers, Playing Card Makers, Plumbers, Poulters
Saddlers (some only, see paragraph 3), Scriveners, Shipwrights,
Tacklehouse and Ticket Porters, Tallow Chandlers, Tin Plate Workers,
Tobacco Pipe Makers, Turners, Tylers and Bricklayers
Watermen and Lightermen, Wax Chandlers, Weavers