High starting torque, synchronous motor, induction motor or DC motor?

It depends on so much more than the simple requirements listed of high starting torque and variable speed. What kind of application are you using it for? Is it on an automobile (where you have DC already), a factory, and do you have the budget and/or space for a variable frequency drive. A synchronous servo motor gives great dynamic control and great starting torque per volume, but its speed range is limited (unless you're field weakening by the back EMF). Servo-motors are also the most expensive due to their position sensors and more intelligent drives.

With a proper soft drive you can go with an induction motor, but it depends. if power is small you can go to step motor also. But dc series motor's starting torque is high as expressed others.
DC Series motors have high starting torque but induction motors have wide range of speed control. So, If DC motor is used, then DC drives you can use, although it will be expensive and DC motors are tough to maintain than ac motors due to commutation Problem.

DC series motor would provide both the high starting torque and adjustable speed BUT beware that DC motors have high maintenance cost and also require AC-DC conversion. You could use other available options e.g. double wound induction motors etc, depending upon your requirements.

But today, there is no application where you cannot apply AC motors, asynchronous or synchronous. If the motor and the associated power electronics are correctly rated, you can have any starting torque you want.

The typical application of DC series motors was in locomotives. This technology has been replaced by AC motors since 20 years. The latest generation of high speed trains use synchronous, permanent magnet motors.