Looking at what I think must be the Beech Midget, Phyllonorycter maestingella (Muller, 1764) on Beech, from which it has rarely been shown to wander. The moth overwinters as a pupa and then flies as the adult in May and June. These lay eggs and the first larval mines appear in July, and the adults fly again in August. These lay eggs again and a second generation of larval mines can be found from late August to October, leaving the pupa to over-winter once more.
The mine tends to be quite long, often from the midrib to near the margin of the leaf. It does turn to be a very tubular mine, with several creases on the underside, which it is said might sometimes be mistaken for one strong crease. There were old mines from the first generation, long vacated.
This is one of the two Stigmella species commonly found on Beech, the Beech Pigmy, Stigmella hemargyrella, which mines towards the midrib, not away from it. This is an old mine, from the first generation of mines of about June. There are several photos on the internet that look exactly like this.
I saw a couple of similar mines today.
On hornbeam, Phyllonorycter esperella possibly and perhaps Stigmella microtheriella.
