I know that the employer of an independent contractor cannot be held liable for negligence of the contractor under vicarious liability, but are there any other such situations where the employer would not be liable for the negligent acts of the employee done within the course of employment?
You do not give a jurisdiction so I will address it from an Australian point of view.
First, you are correct in saying that a principal is not responsible for the tortuous acts of an independent contractor, however, there are circumstances where the principal can be held responsible for the acts of the contractor. Some examples:
Where the contractor is legally the agent of the principal. For example an insurance broker for a general insurance policy (car, building, public liability etc.) is legally an agent of the customer. Conversely, an insurance broker for life cover (life, TPD, income protection etc.) is legally the agent of the insurer.
Where the contractor is an employee at common law. Many arrangements with "independent" contractors are actually sham contracts designed to minimize employer obligations: if the "independent" contractor is truly an employee then the principal remains vicariously liable. Essentially because there is no independent contractor.
Second, an employer is responsible for the tortuous acts of their employees committed in the course of their employment. However, they are not liable if the employee is acting outside the scope of their employment on a "frolic of their own".
This is not a get out of jail free card for the employer, however. If, to the injured party, it appears that the employee was acting within the scope of their employment then the employer remains liable even if the employee was acting outside it or even breaching their contract of employment.
If the employee was breaching their employment contract then the employer can seek indemnity from the employee even though they still remain liable to the injured party. Unless the employment contract is in the jurisdiction of New South Wales, South Australia or the Australian Capital Territory: those jurisdictions have statutes that require "serious and willful misconduct" before the employer can seek indemnity from the employee, breach of contract is not sufficient.
In any event, Commonwealth law prevents an insurer seeking subrogation of a policy against an employee.
An important exception to vicarious liability in the United States would be liability for civil rights violations.
Generally speaking, a governmental employer is not vicariously liable for civil rights violations by an employee unless those violations were carried out pursuant to a written or unwritten policy of the employer, or were a result of the employers negligence in failing to train its employees.
Of course, strictly speaking, since civil rights violations are intentional acts, that would probably be beyond the scope of your question which was directed only at cases of negligence.
Another circumstance would be when there is governmental immunity for claims of the type asserted. For example, in Colorado, the Colorado governmental immunity act sets forth when a state or local governmental may be held liable for negligence. This statute, for example, eliminates an employer's default duty to indemnify its employees if the employee conduct upon which a negligence suit was brought was "willful and wanton" conduct.
Vicarious liability for negligence is also often limited in medical malpractice cases where, for example, the lead medical doctors performing surgery may have vicarious liability for his underlings, but a hospital employing that doctor if the doctor is employed by the hospital, does not have vicarious liability.
The question of vicarious liability can also be complex in the case of a "leased employee" where the employee is formally treated as the employee of an employment agency for HR purposes, but operationally reports on the job site to another firm that pays the employment agency on a per hour or per day rate for the used of the employees services on a long term basis.