I'm not going to add any analysis, but just some RFC quotes and timelines about RSA signing ('signature scheme with appendix'), PKCS #1 v1.5 and PSS ('probabilistic signature scheme')
February 2003 Users encouraged to move away from RSASSA PKCS #1 v1.5 in RFC 3447 PKCS #1: RSA v2.1, §8: Signature schemes with appendix, p. 36
Two signature schemes with appendix are specified in this document: RSASSA-PSS and RSASSA-PKCS1-v1_5. Although no attacks are known against RSASSA-PKCS1-v1_5, in the interest of increased robustness, RSASSA-PSS is recommended for eventual adoption in new applications. RSASSA-PKCS1-v1_5 is included for compatibility with existing applications, and while still appropriate for new applications, a gradual transition to RSASSA-PSS is encouraged. [emph. added]
November 2016 RSASSA PKCS #1 v1.5 is deprecated by RFC 8017 PKCS #1: RSA v2.2, §8: Signature Scheme with Appendix, p. 32, and PSS is declared "required".
Two signature schemes with appendix are specified in this document: RSASSA-PSS and RSASSA-PKCS1-v1_5. Although no attacks are known against RSASSA-PKCS1-v1_5, in the interest of increased robustness, RSASSA-PSS is REQUIRED in new applications. RSASSA-PKCS1-v1_5 is included only for compatibility with existing applications.
Not signing, but encryption*
*I've mixed up RSASSA-PKCS1-v1.5 and RSAES-PKCS1-v1.5 a few times, maybe others have. This is included for their benefit...
PKCS #1 v1.5 for encryption has been deprecated for 20 years.
October 1998 RSAES PKCS #1 v1.5 is deprecated in RFC 2437 PKCS #1 RSA v2.0, §7 Encryption Schemes, p. 11 in lieu of RSAES-OAEP. (same wording repeated in RFC 3447)
RSAES-OAEP is recommended for new applications; RSAES-PKCS1-v1_5 is included only for compatibility with existing applications, and is not recommended for new applications.
November 2016 RFC 8017, §8: Encryption Schemes, p. 17 brings some slightly stronger wording.
RSAES-OAEP is REQUIRED to be supported for new applications; RSAES-PKCS1-v1_5 is included only for compatibility with existing applications.