Remember RSA is just very simple maths, done with huge numbers. If you pick the right "huge" numbers things that look hard become very easy indeed. So we need to ensure we never pick them.
The correct way to do this, which a lot of systems haven't adopted yet, is called RSA-PSS, the Probabilistic Signature Scheme. PSS has a proof that says if you believe RSA works, and assume certain other reasonable things, this is actually safe.
Before RSA-PSS (and still today in lots of backwards compatible systems), people used PKCS#1 v1.5 which has a scheme somebody threw together to do some padding but without any great insight. There is no security proof for PKCS#1 v1.5, it's probably safe, ish, but we can't be sure.
As long as you avoid the known problems, it probably is safe for signatures and the main problem now is that PSS is not included in lots of standards which thus require PKCS1_1v5. This prevents major implementation due to those standards not being updated fast enough.
As an example of slow adoption: The HSMs i'm currently using only started native support for PSS last year, about 20 years after it's introduction.
Please note that pkcs1_1v5 is never secure for encryption/decryption schemes.
More typically to sign a message, RSA is used to sign a hash of the message.
If you have (international) standards to adhere to, you are out of luck most of the time since they specify the exact schemes and cryptography required to adhere to the standard. Adoption of encryption/signature schemes is slow at best unfortunately.
If you do not; go wild. If you like big keys, get some 'quantum proof' public cryptography while you are at it.
What do you suggest here, and will it work with X.509 certificates?
For the Web PKI, the Baseline Requirements currently permit NIST P-256, P-384, or P-521 [sic] for "Elliptic curve" public key signatures, so that would let you do this for "SSL certificates" and plenty of people do but it's not compatible with older software, so if you care about that you need to have a plan B.
Depending your exact browser etcetera, if you go to google.com the certificate you're sent will be one of their P-256 certificates and your browser will verify both that this cert is genuine, and that the server can prove it knows the corresponding private key, using elliptic curve cryptography rather than RSA.