Cantrips are minor (0-level) spells primarily studied by wizards during their apprenticeships. All of these elementary spells take up just one slot in a mage's spellbook, and they are never forgotten, can not be forgotten, any more than a mathematician forgets how to add and subtract just because he is capable of understanding and utilising algebraic topology.

Just as in the real world many jobs require some understanding of maths, from the sales clerk measuring out fabric and counting change (a cantrip level skill) to Stephen Hawkings (at least a 20th level mage), in like fashion, in my worlds people from all walks of life--not just mages-- know and use one or several cantrips; they are woven into the tapestry of society, especially commerce. It's a rare seamstress or tailor who can't cast Stitch, Tie, and Untangle, barbers and hairdressers master Hairy, Untangle, Braid, Depilate, and Colour, housewives use Clean and Polish, and so forth.

After their adventuring career begins, mages may, of course, continue to learn cantrips (from the list below; these are well known and available in any decent centre of arcane study) or devise one of their own creation (subject to GM approval). To learn a cantrip requires 4 hours. As with higher level spells, none of these 4 hours need be continuous; however, time is only recorded in 15 minute increments, rounded down. For example, if a mage studies for 12 minutes, he has barely started to focus and learns nothing; if he studies for 50 minutes, he is credited with .75/hr.

Researching a brand new cantrip takes no less than five days, during which time they must focus on their studies, and they still need access to a decent laboratory and library; they may not research or learn other spells, nor engage in the manufacture or recharging of potions, scrolls, or other magic items, during that time (although participating in the normal activities of daily living is fine, other than drinking to excess or using hallucinogenic drugs). The basic cost of research = 100 -- 1,000 gp (1d10) per spell level; a cantrip requires ¹/₁₀ as much, so 10 -- 100 gp. If the spellcaster meets all the expenses and spends five days in dedicated research, that evening he may attempt a success roll. The chance of success is 10% and is modified by the caster's intelligence score and experience level.
Day 5: basic success chance = 10% + 1% per point of relevant ability score + 1% per experience level
If the first attempt is unsuccessful, research continues . At the end of each additional day a new check is made with a +10% cumulative bonus.
Day 7: basic success chance = 10% + 1% per point of relevant ability score + 1% per experience level + 20% bonus
However, a roll result of 99 or 100 on any success check means that the cantrip is unworkable in some fundamental way and must be abandoned. The mage may start over, trying a different approach to achieve the same result, but that requires a fresh investment of both time and money.