Checkpoint.

Is Singular capable of checkpoint? Is singular capable of saving a state of calculations, and returning there rather than at the very beginning, in the case of computer crash? If yes, how can I do that?

Suppose you mean by checkpoint what is explained in wikipedia under Application checkpointing:
https://en.wikipedia.org/wiki/Application_checkpointing
Check whether Singular's dunp command https://www.singular.uni-kl.de/Manual/latest/sing_227.htm#SEC267 can help you.