What is important is to differentiate people who are still in the asylum process and those whose application procedure was terminated negatively. In the first case you have to apply for a training permit only.
1. You can apply for an Ausbildungsduldung if the asylum procedure ended negatively, not during the procedure.
2. Whether you need a Duldung before or not depends on the circumstances of the individual case. To obtain an Ausbildungsduldung according to § 60c Residence act, there are several prerequisites you have to comply with:
1. Duldung according § 60a Residence act before you apply for an Ausbildungsduldung for at least three months (This doesn't apply to persons with a so called "Duldung light" acc. to § 60 b Residence Act - they cannot receive a training permit at all)
You don't have to meet this condition if you came to Germany before 31.12.2016 and start your apprenticeship before 1.10.2020.
2. Clarified dentity, There are diferent deadlines, depending on when the person in question arrived in Germany. If he or she came before 31.12.2016, the identitiy has to be clarified until they start their apprenticeship (1.9.2020 as a rule), if they arrived after 1.1.2017, but before 31.12.2019, the identity must be clarified until 30.06.2020. All newcomers will have to clarify their identiy within the first six months after their entry into Germany.
3. No offences above 50 or 90 penalty units - 50 is for criminal offences /90 for offences concerning Asylum/Residence law.
4. No imminent measures of termination of residence.
This implies that an application for an Ausbildungsduldung should be sent to the immigration authoritiy immediately after the final decision of the court is legally binding, so that the immigration office won't activate the deportation process.