Restrict To Z

Axis Constraints toolbar > Restrict to Z

Restrict To Z lets you limit all transformations (move, rotate, scale) to the Z-axis.

When you click Select And Move and then click Restrict To Z, you will be able to move the object only on this axis. Use this when you want to make sure you don’t inadvertently move an object in any other direction.

You can also restrict transformations to the Z-axis when using the Transform gizmo. Selecting the Z tripod head will allow you to move only in Z.

Note that constraints are set on a transform-by-transform basis, so select the transform before you select the axis constraint. If you do not want the constraints to change, turn on Customize menu > Preferences > General tab > Reference Coordinate System group > Constant.

The axis constraints are stored separately at object and sub-object levels. If you set these three controls one way while in sub-object mode and another way while in object selection level, when you return to sub-object mode, they're restored to the way they were previously set. For example, if you're using XY constraints at object level, then switch to sub-object level and use Z constraint, when you return to object level, XY will be restored.

Snaps can conflict with Restrict To Z. Go to Customize menu > Grid And Snap Settings > Options > Translation to find the Use Axis Constraints check box. When this is off, Snaps take precedence over axis constraints.