Patterns flaw to sign a counter with public key encryption protocol In modern asymmetric encryption protocol, like secp256k1, are there any flaws if you want to sign a counter? 
i.e: use a private key to sign from 1-n is that any chance make n+1 easier to guess?
Terminology issues:


one does not sign with an encryption protocol, but with a (digital) signature scheme.
secp256k1 is neither of these. It is an elliptic curve.


Thus it is wanted to ask if in modern (asymmetric) (digital) signature schemes, like ECDSA on elliptic curve secp256k1, there is any flaw in signing a counter.

The answer is no, for proper implementation. That would go against the theoretical definition of a signature scheme's security. And there is practical insurance that the scheme does not become insecure when a counter is used as message: in all modern signature schemes with appendix (including ECDSA), the message is hashed, the resulting hash is the only input to the rest of the signature scheme, and the hash of a counter has no remarkable property (for a secure hash).

The only remarkable things are that


The messages are predictable. It allows the signer to prepare the signatures in advance. If either comes with a risk, it is not one specific to the signature scheme.
The messages are heavily constrained, which can only raise the difficulty of signature forgery, compared to other predictable messages.