Sure you can have a horse that's homozygous solid...they just don't carry any white genes.
I'm assuming you mean white patterns. I'm not entirely knowledgeable on how tobiano works if it causes different markings between hetero/homozygous (other than it isn't uncommon for homozygous to have "paw prints", I believe). Though it is entirely possible to have a very minimally marked tobiano.
Obviously you cannot have a homozygous frame , ha. As for splash and the test becoming recently available, it appears to cause minimal markings when heterozygous and over 50% white on homozygous (see the examples here: Morgan Colors- Splashed White Morgans
As for sabino, rabicano, etc, I am unsure.