Premier and Caravan - unless it's required by the regulations (i.e. 135/121) then no.
Important to note - Some light business jets are certified for single pilot operation. Others are not certified for single pilot operation, but can be flown by a single pilot, provided *HE* has an extra single-pilot checkride and goes for a single-pilot checkride every year. (I believe this type of setup is done under an FAA waiver or exemption of some kind). The best thing to do to figure out if a certain airplane is single-pilot certified is to take a look in the Limitations section of the Airplane Flight Manual, or look at the Type Certificate Data Sheet for that aircraft here:
If the TCDS says, under Minimum Crew, that there must be two pilots, for all flights, then you may log SIC, regardless of what checkrides the PIC may or may not have done. In all cases, you may log SIC if it is required by the type of operatoin being conducted (such as 135/121 et al)
P.S. Logging of SIC can only be done in accordance with 14 CFR 61.55.