Please read the attached document. Serine proteases and carboxypeptidase A in conjunction with zinc are very relevant in the cleavage reaction but could also occurArticle Spontaneous cleavage of proteins at serine and threonine is ...
Article Specific cleavage of amino side chains of serine and threoni...
The MEROPS database is a rich source of information about proteases/peptidases.
https://www.ebi.ac.uk/merops/index.shtml
As far as I can tell, there is no protease that selectively cleaves C-terminal or N-terminal to Ser or Thr irrespective of the surrounding residues, the way trypsin cleaves at basic residues.
As stated by Adam there is no protease specific for serine but selective hydrolysis of peptide bond immediatly downstream of serine(and threonine) is possible using zinc... But it is slow.
Article Spontaneous cleavage of proteins at serine and threonine is ...