Does anyone know whether it is common for CAs to issue end entity certificates, which can be used as client certs (possibly among other purposes), with the Extended Key Usage (EKU) extension marked as critical? I think that most CAs do not mark EKU as critical in the certs that they issue, but I would like to have my assumption confirmed.
While absence of evidence isn't evidence of absence, the general answer is "no, TLS certificates do not get their EKU marked as critical".
- RFC 5280 says "This extension MAY, at the option of the certificate issuer, be either critical or non-critical."
- CA/Browser Forum's Baseline Requirements v1.6.3 has opinions regarding root EKU (don't have them) and intermediate EKU (if needed, SHOULD be non-critical); but on end-entity (or "subscriber" in their terminology) EKU it has no opinion regarding criticality.
- For timestamp authorities RFC 3161 says "This extension MUST be critical."
So, critical EKUs exist, but it's not common to have critical EKU for a TLS Server or TLS Client certificate.