Your setup should not need STP but it is there and as the administrator, you must protect your network from the "morons" as Tom mentioned above.
There are also some network attacks that will bring down your network...
Usually you would want to keep STP and it's modern variants (RSTP, MSTP) away from the core(s) and configure your distribution switches as your designated spanning-tree primary and secondary roots.
Note:- Your distribution switches appear to be access switches and I don't think they would support an IGP routing protocol such as OSPF to connect to your core switch.
This leaves your only core switch (spanning-tree root) subject to whatever nightmares your clients will bring to the network... and you don't want that phone call on a Friday afternoon telling you the network is down while you should be going fishing for the weekend!!
There is a bit of work to do to protect your clients from themselves and you may have to do it in approved Change Management windows (ITSM)
At the root trunks investigate "spanning-tree guard root" and other features such as UDLD. https://supportforums.cisco.com/discussion/10707896/spanning-tree-guard-root-command
On the access switches at the switchports, possibly plan to roll out port security and if required "IP DHCP Snooping" and possibly DAI http://packetpushers.net/yes-we-really-need-dynamic-arp-inspection/
If not already, use a syslog server (some Linux virtual machine) to log what happens on your network for if you have to investigate/fix what caused a network outage, it will help.