SpaceX has re-used a first stage once, several times now. Has it re-used the same first stage twice yet? If not, has a particular mission been named that will be the first second re-use?
Yes. As of December 3rd, 2018, SpaceX successfully launched the same first stage (B1046) 3 times and retrieved it twice. It supported the Bangabandhu-1 mission in May 2018, Merah Putih mission in August (setting the current record for fastest turnaround time of 3 months), and then the SmallSat Express mission on December 3rd 2018.
No. The cores used are tracked at Reddit, among other places. There are no plans to reuse a booster for a third time. There are 3 booster now that have been reused
- B1021- Will be displayed at cape, retired.
- B1029- Likely mothballed.
- B1031- This one just landed today, not sure of it's fate yet.
It will likely be a while before a second reuse is done, but they are slowly inching forward in that direction. I would guess one of the Block IVs, maybe B1042.
The current plan is to only reuse more then once the Block 5s. Currently SpaceX is working on expending all of the Block 3 and 4s in existence, to focus on Block 5, for a number of reasons. One of the most important is NASA wants to use Block 5 for the crewed missions, and needs 7 successful missions in a row before they will allow it to be used for manned missions.