Simon Fishwick said he was concerned that the materials were not effective enough in tackling the spread of the virus and suggested they were becoming more of a "fashion accessory".
Councillor Fishwick runs Earnshaw's greengrocers in Horbury and represents the town for the Conservatives on Wakefield Council.
He told a public discussion about the pandemic that he was "confused" by the guidance around face coverings.
The government made masks mandatory on public transport and in shops last month.
Speaking at a health scrutiny meeting on Thursday, Coun Fishwick said: "I run a shop, but I've got people coming in using socks, underwear garments, shirt sleeves and scarves.
"I'm completely not getting this.
"If you've people dealing with asbestos they've got certified masks. What we're seeing is more of a fashion accessory. I'm seeing masks with Leeds United and Castleford Tigers (on them).
"What I'm seeing in Horbury and surrounding villages is face masks being discarded all down the streets.
"If it's as bad as what we're being told why aren't these masks being disposed of in a bio-hazardous box, collected by the council and incinerated."
Stephen Turnbull, from Wakefield's public health team, said the science behind the virus meant that face coverings were effective, but he added that using a single sock "probably isn't helpful".
He added: "It's better if a face covering is at least double layered. There's guidance on the government website about the sort of standards you need.
"The virus is invisible but it can only transmit where there's a droplet.
"The point of a face covering is it won't cut out those smaller particles, but it will cut out the big ones.
"It doesn't protect the person who's wearing it as much as those near them.
"If you're breathing out or you're talking, there's lots of droplets. If you've got Covid that face covering will stop them.
"It will probably stop the person you're talking to becoming infected."
Addressing the issue of the masks being discarded on the ground, Mr Turnbull added: "It annoys me greatly to see litter and waste anywhere.
"It's a by-product of the age we live in unfortunately and we don't like to see it."
Local Democracy Reporting Service