According to RFC 3447 OAEP padding scheme can be used only with encryption operation and therefore it cannot be used with signatures. If standard RSASSA-PKCS1-v1_5 scheme is not good enough for you I believe you should use RSASSA-PSS which is recommended for new applications and have characteristics similar to OAEP scheme. See RFC 3447 for more details.
You can easily check out that my answer is correct with command line OpenSSL tool:
Generate private key:
openssl genrsa -out private.key 2048
Generate some input data:
echo "Hello world" > input.data
Try to generate signature with OAEP scheme:
openssl rsautl -sign -oaep -inkey private.key -in input.data -out output.data RSA operation error 139655304349344:error:04066076:rsa routines:RSA_EAY_PRIVATE_ENCRYPT:unknown padding type:rsa_eay.c:389:
Try to encrypt data with OAEP scheme:
openssl rsautl -encrypt -oaep -inkey private.key -in input.data -out output.data
You can also take a look at "rsautl" module source code if you need more information but remember that RFC 3347 is your friend :)