What are the recommended approaches to Cloud Software Security?

Reiterating few and adding more to what is mentioned earlier. Some of these apply to all software and not just those running on cloud

1. Setup firewal policies carefully. Only open ports which are absolutely necessary. Blindly opening all ports is easy but unsecure.
2. Based on the cloud provider select try to go for a vitrual private cloud so and allow access to the nodes over a VPN only.
3. Encrypt traffic to/from outside the cloud using HTTPS/SSL.
4. Use an encrypted file system. Some cloud providers like Amazon advise this. Encrypt user information in database for very sensitive data.
5. Proactively monitor and manage the system with security patches, upgrades, etc
6. Use strong password policies. For connecting to servers over ssh, etc use key based access as opposed to password based login.
7. Use a third party PCI compliant system for storing critical information like users's credit cards.
8. Do an OWASP(https://www.owasp.org/index.php/Main_Page) security vulnerability check on your application.
9. Select a cloud provider which has a good infrastructure to detect intrusion and also physically secure and certified in security industry standards.

If you talking about Azure cloude service it has there own security.

1. Fence the system : Use of firewalls, packet filtering on nodes/devices facing the public. No nodes should be facing public except a reverse proxy which will pass the traffic to the front ends then to the application layers.

2. Use integrity checking software such as tripwire or rootkithunter 

3. If possible use Antivirus Gateway as part of the public facing node.

4.  Implement good auditing. Use tools such as splunk for logs inspection with proper log indexer and probably a dedicated node.

5. Have a disaster recovery plan - backups strategy, HA strategy. 

In short its. Deter, Prevent, Detect , Correct.