It depends.
For instance, an action game that you have a mini game inside that you run to save yourself from hunters, might need only checkpioint saves.
You might also need to make a horror setting. A feeling that you know what? You are not safe. You are actually freaked up (rephrase that).
So you need small rooms with checkpoints of even one use!
Like resident evil with typewriters and ink to be used to save.
On the other hand, an RPG with an open world could be suitable to save everywhere.
Save everywhere or save in certain times can be used combined too.
What you have to be careful, is exploits. If a gamer can save the game and try stuff making it easy as pie, the game WILL be exploited and it will become boring soon enough to not worth the time to be played. If the game won't be broken by overusing save feature, it's ok to save everywhere.
If this... bends the rules of the game it's a bad thing. Here is an example.
You wannt a gamer to answer correct to 5 questions.
The gamer must do 5 dungeon runs to know the right answer.
There are four possible questions.
Player saves.
"Ask me the questions Bridge Keeper I am not afraid!"
Question 1 wrong. Died.
Load.
Question 1 wrong. Died.
Load.
Question 1 correct.
Save.
Question 2 wrong. Died
Load.
Question 2...
See what happens here? Trial and error breaks the riddle solutions.
This sould NOT had a save.
This should be a scenario like: Do something a LITTLE boring, then find the man with questions. Complete all 5 puzzles, learn the answers, go and say the correct ones to GO ON AND SAVE. Or else, go back before the LITTLE boring process and do it AGAIN!