If you originally signed up with Facebook but:

• You no longer want to sign in through the app, or
• You deleted your Facebook account linked to SoundCloud

You can still access your SoundCloud account by setting a new password. 

Request a password reset link here using your profile URL or the email linked to your Facebook account. The reset email will be sent to your Facebook-associated email. If you don't see it, check your spam folder.
After resetting your password, you can log in with your email and password instead of Facebook.

To fully disconnect Facebook, log in and do so here.
 

I don't know the email address attached to my Facebook account.

If you still have access to your Facebook account, you can view your attached email by going to your Facebook settings.