There is no direct access to the layered nav from the SOAP and REST api, and you cannot check which attributes can be used in the layered nav with the default install. Take for example the catalogAttributeEntity
from the WSDL. It only returns a few details about the attributes.
<complexType name="catalogAttributeEntity">
<all>
<element name="attribute_id" type="xsd:int" minOccurs="0"/>
<element name="code" type="xsd:string" minOccurs="0"/>
<element name="type" type="xsd:string" minOccurs="0"/>
<element name="required" type="xsd:string" minOccurs="0"/>
<element name="scope" type="xsd:string" minOccurs="0"/>
</all>
</complexType>
You can try to extend this functionality to give you only the filtrable attributes and based on category or search results, but It's a long process.
if you manage to implement this, the rest should be easy, because you can supply filters to any product collection with SOAP or REST calls.
See this as example for soap: http://www.magentocommerce.com/api/soap/catalog/catalogProduct/catalog_product.list.html
and this for rest: http://www.magentocommerce.com/api/rest/introduction.html#RESTAPIIntroduction-Products
So to answer you question, The apis do not fully support layered nav but it can be achieved with some code.