In the US the closest thing to a standard, and the requirement (per the FAA's practical test standards), is that you make appropriate use of checklists.
As mentioned in that video "appropriate" leaves some room for common sense interpretation (clearly your first course of action if your engine dies in flight should not be to open the operating manual to the Engine Failure In Flight checklist and begin reading: That would NOT be "appropriate").
I expect most other countries have a similar view on the subject.
As far as what gets memorized, the checklists for most simple aircraft (for our purposes let's say that's anything with a single piston engine, unpressurized, with 6 seats or less -- your average Piper/Cessna/Citabria/etc.) can be memorized. They're not that long and there's not that many of them.
Time permitting a good pilot will generally still refer to the printed checklist even if they did the entire thing from memory, if only to read off each item and make sure they didn't miss anything.
More complicated checklists (like what you might find on a 737 or a C-130 for example) will usually have a set of Immediate Action Items (often called "Red Box Items" because many manufacturers put them inside a big red box at the top of Emergency or Abnormal checklists).
It's generally expected that the flight crew have the Red Box Items committed to memory and be able to complete essentially on instinct.
After completing the Red Box Items all the really important stuff is done, and there is hopefully enough time that it's now appropriate to pull out the checklist (or Quick Reference Handbook), verify that the Red Box Items have all been done, and run through the rest of the items.