As a structural engineer, we offer 2 levels of observation services on our projects.
The first one is typically included in the fee,(but still calculated into the hours on which the fee is based), and covers limited observations of field work to see that things are being carried out "in general conformance" with the design intent. Depending on project size and complexity, this typically involves 2-5 site visits during the construction phase.
The second one is often called "special inspection" in our area, and we consider the work as "additional services", and charge a separate fee. This requires being much more involved in the construction observation of our part of the project, e.g. counting reinforcing bars in footings and slabs, inspecting formwork for proper depth and clearances, watching concrete pours or grouting of masonry walls, etc.
Other engineers, such as Soils Engineers, typically ask for the second type of service on most of their jobs, because their recommendations depend highly on how the Contractor performs their work. In our case, we usually request that we perform it only on more complicated projects, where the potential for the Contractor to screw things up is higher.
My own opinion is that the level 2 type of service probably wouldn't be warranted on a residential type of job, (unless, of course, it was my house getting built).