Full Idea
The general view is that Russell's Paradox put paid to Frege's logicist attempt, and Russell's own attempt is vitiated by the non-logical character of his axioms (esp. Infinity), and by the incompleteness theorems of Gödel. But these are bad reasons.
Gist of Idea
The standard objections are Russell's Paradox, non-logical axioms, and Gödel's theorems
Source
Crispin Wright (Frege's Concept of Numbers as Objects [1983], 4.xvi)
Book Reference
Wright,Crispin: 'Frege's Conception of Numbers' [Scots Philosophical Monographs 1983], p.131
A Reaction
Wright's work is the famous modern attempt to reestablish logicism, in the face of these objections.
Related Idea
Idea 13863 Logicism seemed to fail by Russell's paradox, Gödel's theorems, and non-logical axioms [Wright,C]