For me, it depends on the house and the inspection results. If the house is newer and looks O.K. I will recommend my clients to get both home inspection and termite inspection (from two different companies); from their recommendation and findings, we will go with specialized professionals when needed.
For older properties, in addition to the two inspections, we might just go ahead and get specialized inspections.
For properties with trees, we will get arborist if the giant heritage oaks are part of reasons why the clients like the property. Pools, pool inspector, unless the pool works great and looks great. Septic system needs special inspection. etc.
Of course, we only recommend and advise, the clients are the ones who will be making an educated decision on what they want to do.
Either way, we recommend companies with great reputation and adequate insurance just in case they make a mistake.