I like this question ...
Weeks ago I placed a similar one at german authority (LBA):
"Is there a list of Annex-I aircraft registered in Germany?"
... No answer, not even a number.

Each individual aircraft must be evaluated and must be checked towards requirements set out in Annex-II to BR.

In aircraft certification (that’s the process to obtain a Type Certificate), no requirements for training aircraft exist. Consequently, it is not possible to certify/approve a 'training aircraft'. Having said this there can't be a list of aircraft approved for training, in consequence.
This does not mean it is not possible to design an aircraft tailored to specific training purpose.
The evaluation whether an aircraft is suitable for a given training envisaged is to be done by each DTO/ATO individually.

What Jannes writes is true for EASA aircraft, but for Annex I aircraft, there actually is an approval requirement. See ORA.ATO.135(a)(1) and (2) and DTO.GEN.240(a)(1) and (2). However, the approval is not done by EASA but by each individual Competent Authority, so there would not be one list, but several.

If I remember correctly; to be able to undertake a Skill Test for a CPL you'll need a variable pitch/constant speed propeller and a retractable undercarriage. Can't seem to find the reference at this point, though.