Synes expresses an opinion based upon feeling:

Jeg synes det er en god bog. (I find that it's a good book - I 've read the book and I like it).

Synes says: That's how I feel about it!

Mene is an opinion based upon conviction. Verifying is not within reach.

Jeg mener det vil være godt for økonomien at sætte skatten ned. (I really do believe lowering of taxes would be good for the economy).

Mene says: Believe me, I'm serious! Or: I'm almost sure!

Choose: tænke, tro, synes or mene in the drop-down-list

Jeg menersynes, at moster Annas sko lugter rigtig, rigtig dårligt

Henne i banken menersynes de, at jeg skal sælge mine Novo-aktier nu. De er begyndt at falde.