TL:DR – most employers will take with a grain of salt any “bad reference” given by a former employer regarding a new hire. Two exceptions to the rule: undisclosed criminal record and untruthfulness.
A few days ago or so, I saw a post on this sub about not giving your employer any notice when you quit. IIRC, they used the example of this model perfect employee who agreed to 4 weeks before quitting. Allegedly, the rogue ex-employer used that extra time to bad mouth the employee to the new employer. The employee was then fired and when they went to the new employer, the new employer fired them as well due to the bad reference from the ex-employer. Ergo, this served as a cautionary tale not to give your employer any notice when you quit or tell them where you go to work, next.
I'm not saying this didn't happen. There are some really messed up employers out there, so who knows. But understand this is the exception to the rule and not the norm. Please give the customary 2-week notice, before quitting — unless doing so is not practicable.
If an ex-employer of a new hire contacts HR and bad mouths the new hire, they will most likely be required to reduce their “reference” to writing. If the ex-employer refuses to put in writing the criticism of the new hire, HR may ask them to give a verbal over-the-phone statement about the new hire, which will be recorded. If they refuse to do that too, none, if any weight, will be placed on the ex-employer's bad reference.
There are however, two exceptions to the rule above:
(a) Criminal record. If the ex-employer discloses that the new hire has a criminal record, which was not disclosed by the new hire, such reference will not have to be in writing or recorded. We can check various law enforcement databases to see if the new hire's name comes up. If it does, and it's for something trivial like small possession of a controlled substance, non-violent domestic complaints, or other non-serious offenses, then it's not likely going to result in withdrawal of the offer. On the other hand, if it's for something serious, like DUIs, attempted-[insert crime], it will most likely result in withdrawal/firing of the new hire.
(b) Not giving/following the 2-week notice. If the new hire states they need to give their 2-week notice to their ex-employer, but then the ex-employer calls to complain that the new hire didn't give any notice, we may try to confirm the veracity of the statement. For ex., we might contact the new hire and ask them to start earlier than when we agreed for them to start. If the new hire refuses on account that they are still working for the ex-employer, we'll give the benefit of the doubt and do nothing else. On the other hand, if the new hire starts earlier without problem, then that's an indicator that the ex-employer was truthful.
To be clear, it's not that the new hire didn't follow through with the 2-week notice that's the problem. Instead, it's the fact that they were untruthful. That's the main problem. That said, we will keep it noted that new hire just walked up and left from past employer and then didn't tell us the truth. This means the new hire will likely be under much more scrutiny (especially on matters dealing with ethics) and their probationary period may be extended.
Should you tell your ex-employer where you found a new job? That depends on your work-relationship with your ex-employer. If it's a bad/toxic one, then the answer is obvious no — you may even want to forgo the 2-week notice –> if you do this though, make sure you tell your new employer you can start right away and don't need to give notice to your ex-employer. Remember it's the truthfulness that counts not necessarily the actions. If the ex-employer then calls and complains about the notice, then we are already aware of the issue. On the other hand, if you had a great working relationship with your ex-employer, you can absolutely tell them where you are going.