There are many conventions for Euler angle sequence, just are there are many conventions for coordinate system axes.
Robotics generally uses right-hand coordinate frame for robots and robot parts, with x-forward, y-left, z-up. Euler angles in robotics are generally ZYZ ; using an a-b-a sequence (ZYZ, YZY, XZX, ...) rather than a-b-c (XYZ, YZX, ZYX, ...) sequence helps avoid singularity/gimbal lock issues.
However, robotics also often uses the optical/cameras/sensor-systems coordinate frame convention of z-forward, y-down, x-right for cameras and some other pixel-wise sensors which project the 3D world to a 2D image/matrix.
For an example of making conventions explicit, see the ROS REP 103 on metrics, coordinate systems, and rotation representations, and REP 105 on coordinate system naming, chaining, and conversion to Earth-centred/global systems and maps.
There are lots of other conventions. For example, see this explanation of the NASA Standard Aeroplane and NASA Standard Aerospace coordinate systems, which are x-fore, y-up, z-right and x-fore, y-up, z-right.
edit: This paper (Rotations in Three-Dimensions:
Euler Angles and Rotation Matrices) gives a clear summary of the Euler angle parameters on pages 1 & 2, gimbal lock on pages 5 & 6, and problems with averaging Euler angles on page 8.