So, today 4.1 hit here in the US. I was not expecting TSM to break in any way as the only substantial change that blizzard made had to do with addons that use the hidden addon chat channel to communicate between players which TSM doesn't do at all. All I have done as of writing this is cancel my auctions, use the "Open All" from the Mailing module, and then repost my auctions. So far I haven't run into any problems with TSM.
However, I have heard of people having issues with not being able to post. This issue is a problem with Auctioneer. The short explanation of why Auctioneer is making TSM not work is that Auctioneer replaces all the AH functions with their own functions. So, if their functions break, TSM is using those broken functions and that'll cause TSM to not work. The solution here is to simply disable Auctioneer while using the Auctioning module of TSM until the developers of Auctioneer fix the issue.
I have also heard of people having issues with not being able to open mail. I could not replicate this issue. However, I don't have any other mailbox addon such as Mail Opener or Postal. If you are using Postal or MailOpener, note that TSM_Mailing will hide its "Open All" button and allow Postal / MailOpener's equivalent button to be shown so if you are having an issue it's likely with Postal / MailOpener.
If you are having any problems at all even with only TSM enabled, please let me know so it can be addressed asap either through IRC or through the official TSM forum.
PS: Make sure you check the "Load out of date addons" checkbox in the addons frame in your character selection screen. In the upcoming days, all TSM modules will be updated so that they no longer show as out of date, but for now don't lose sleep over it saying that TSM is out of date.