The existential 'there' is usually followed by a form of the verb 'to be', used as a pure copula.
For instance, rather than saying,
a wrench is on the bench, you'd say
there's a wrench on the bench.
Biber, Conrad and Leech (2002) list a number of syncategorematic/copular verbs which can also follow the existential 'there', eg:
There used to be a house … there's supposed to be a plot … there seems to have been a mistake … there's said to be a ghost …
They don't make a distinction which I've noticed about what I call 'emergent-type' verbs, always used in the past, describing, as I see it, an action which has emerged, but seems as yet incomplete, eg:
…there arose such as clatter … there appeared a great multitude … there emerged, out of the freshly tilled soil, …
I noticed today another type which I hadn't clocked before … the use of the intransitive after 'Once upon a time', eg:
Once upon a time, there lived …
I can't think of an instance where you'd use a transitive verb in this construction.
Are there any other types of verbs that can follow the existential 'there' in declarative sentences that you know of that I can add to my list?