The core branches are:

Mechanical Engineering, Civil Engineering, Electronics Engineering and Electrical Engineering in most cases chemical engineering is also consider as core engineering.

The branches which are easily available in every college's are:

Mechanical Engineering

Civil Engineering

Computer science Engineering

Electronics and Telecommunication Engineering

Electronics engineering

Electronics and Communication Engineering

IT Engineering

Electrical Engineering

EEE engineering

And other Engineering courses are:

Software Engineering

A.I Engineering

Biotechnology Engineering

Aerospace Engineering

Aeronautical engineering

Chemical Engineering

Petroleum Engineering

Instrumentation Engineering

Production Engineering and there are many more.

Dear aspirant,

There are four core branches in Engineering,

Civil engineering.

Mechanical engineering.

Electrical engineering.

Electronics and Communication Engineering.

These four branches are considered because these are base of government sector.Apart from this there many non-core branches in Engineering.

Hope this helps!