An apprentice/trainee does not have a statutory right to claim appointment even after successful completion of apprenticeship. The employer is not under any statutory obligation to give him employment.
However, if the terms of the contract of apprenticeship lay down a condition that on successful completion of apprenticeship, the employer would offer him employment, then it is obligatory on the part of the employer to do so.
Similarly, the period spent on apprenticeship/training is not ordinarily countable towards regular service. However, if the terms of apprenticeship provide otherwise, then the employee is entitled to claim it as part of service. The relevant case-law on this subject is Haryana Power Generation Corp Ltd v. Harkesh Chand, 2013(1) SCC L&S 387