Your signature is not a private key in any sense if the phrase.
It's pretty much as public as it gets!
You publish it to everyone that cares to look for it by the act of writing it on the cheque.
Written signatures are also known to be relatively simple to fake to a high enough standard to fool a bank, and if given enough time, to fool even the most detailed scrutiny.
A certified signature is needed for some deals - A solicitor trusted by both parties takes a record of the document and can later be asked to confirm whether the document is unchanged and the right person signed it.
That one is similar to "signed executables" or signed drivers, except different because those are only saying "Hasn't changed since signing" as the trusted external party has never seen the document.
You have to fully trust the signer to be nice - and to look after their private keys.