Flap endonucleases (FENs), essential for DNA replication and repair, recognize and remove RNA or DNA 5′-flaps. Related to FEN specificity for substrates with free 5′-ends, but controversial, is the role of the helical arch observed in varying conformations in substrate-free FEN structures. Conflicting models suggest either 5′-flaps thread through the arch, which when structured can only accommodate single-stranded (ss) DNA, or the arch acts as a clamp. Here we show that free 5′-termini are selected using a disorder-thread-order mechanism. Adding short duplexes to 5′-flaps or 3′-streptavidin does not markedly impair the FEN reaction. In contrast, reactions of 5′-streptavidin substrates are drastically slowed. However, when added to premixed FEN and 5′-biotinylated substrate, streptavidin is not inhibitory and complexes persist after challenge with unlabelled competitor substrate, regardless of flap length or the presence of a short duplex. Cross-linked flap duplexes that cannot thread through the structured arch react at modestly reduced rate, ruling out mechanisms involving resolution of secondary structure. Combined results explain how FEN avoids cutting template DNA between Okazaki fragments and link local FEN folding to catalysis and specificity: the arch is disordered when flaps are threaded to confer specificity for free 5′-ends, with subsequent ordering of the arch to catalyze hydrolysis.