The Seasiders were looking to give their survival bid a major boost but could not make a breakthrough against the Potters, for whom Kenwyne Jones missed the best chance of the game when he rounded goalkeeper Matt Gilks but shot wide in first-half stoppage time.
Glenn Whelan also went close for the visitors, while Asmir Begovic did well to deny Matt Phillips from close range.
Blackpool had scored and conceded in all their previous top flight home matches this season - but this blank means they remain 17th in the table.
Home boss Ian Holloway named an unchanged team, while Potters boss Tony Pulis made one enforced adjustment from the 3-0 victory over Wolves on Tuesday, replacing hamstring injury victim Matthew Etherington with Dean Whitehead.
Blackpool carved out the first goalscoring opportunity after six minutes, Stephen Crainey whipping in a cross and DJ Campbell heading over the bar.
Two minutes later Phillips collected a loose ball and tried his luck from outside the area, but skewed his shot wide.
Soon after Charlie Adam attempted to burst through and collided with Whelan and Robert Huth, with the latter's foot appearing to make contact with the Blackpool captain's head. Adam looked to be in some pain, but was able to continue.
Having soaked up some pressure, Stoke got their first strike in on goal as Huth saw a shot blocked by Crainey, but the Potters defence was then almost caught napping when a long ball sailed over their heads, Marc Wilson just on hand to make sure Phillips could not take advantage.
Stoke broke forward again with Rory Delap delivering a cross into the danger zone, but Alex Baptiste did enough to avert the danger.
Neal Eardley sliced well off-target having been teed up in the box before his defensive colleague Ian Evatt made a great tackle at the other end to stop Jones advancing.
Blackpool then won a free-kick in a useful area following Whelan's foul on David Vaughan, but Adam could only find the Stoke wall with the effort.
Gary Taylor-Fletcher got on the end of a cross from Evatt four minutes before the interval but was unable to get much power behind his header, which Begovic gathered comfortably.
The hosts then survived a real scare in first half stoppage-time when they lost possession and the ball came to Jones, who took it around Gilks as Evatt slipped over.
It seemed like a simple finish for the former Sunderland striker, but he somehow contrived to blaze wide and the score remained 0-0.
Phillips went close at the start of the second half but his flick towards goal from Adam's corner was saved by Begovic.
Keith Southern then fired an effort in which took a deflection off his team-mate Campbell and went narrowly wide.
Stoke responded with a swerving shot from Whelan which Gilks dived to palm away and a downward header from Jones that was scrambled clear, while Pulis threw on Ryan Shotton for Andy Wilkinson.
Blackpool fans appealed for a penalty when Campbell got in a tangle with Whelan in the box, but referee Mark Clattenburg waved play on, and Vaughan tried a drive from distance which flew over.
Holloway refreshed his attack with just under 20 minutes to go, bringing on Luke Varney and Sergei Kornilenko for Phillips and Taylor-Fletcher.
Pulis made another substitution of his own, introducing John Carew for Delap, before Whelan sent a fierce shot wide.
Kornilenko got a header in towards the end that did little to trouble Begovic as Stoke held on to frustrate their opponents.
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