After going 10 games unbeaten up until 15 December, the second best record in European football at the time, Norwich have gone nine league games without a win and been knocked out of the FA Cup by non-league Luton.
But the two sides are both on 29 points after a result that edges them ever so slightly closer to Premier League safety.
Both teams extend the gap from the relegation places to eight points, although Norwich boss Chris Hughton may have greater cause for concern.
After going 10 games unbeaten before Christmas, they are now on a winless streak almost as long.
Their predicament has been caused mainly by a lack of firepower, having scored just once in their last six matches.
Norwich need more guile - Hughton
Argentine striker Becchio was signed in January to improve the Canaries' goalscoring form, and an early chance fell his way, but his shot from 10 yards was easily dealt with by Schwarzer.
As it transpired, the former Leeds man wouldn't have a better opportunity all afternoon.
The only other notable strike of a dire first half was a high kick from Bradley Johnson that hit Dimitar Berbatov in the forehead, forcing him temporarily off the field for stitches.
The elegant Bulgarian seemed more offended by some of the efforts of his own team-mates, whose sloppy passing - especially from defence - curtailed most attacking forays before they had caused any damage.
Steve WilsonBBC Match of the Day commentator
Luciano Becchio's early chance proved to be the closest we came to a goal. Real opportunities were few and far between at either end, but it's a valuable point for both sides.
Fulham began with greater purpose after the break, Sidwell forcing their first shot on target with a powerful strike that was parried by Mark Bunn.
But Norwich weathered the small storm and the match soon looked every bit a contest between two out-of-form teams, at times resembling a war of attrition.
Elliott Bennett and Robert Snodgrass huffed and puffed trying to create something for the hosts, but with Becchio comfortably marshalled by Fulham's experienced centre-backs, clear openings were rare.
Fulham controlled the game - Jol
Norwich finished the stronger, with Elliott Bennett having a shot blocked and Johnson scuffing a chance wide, but the final 0-0 scoreline was a fair reflection of the game.
Norwich manager Chris Hughton:
"Fulham have a lot of quality in their side, but we are disappointed because we are at home and the emphasis was to look to get three points.
"We did not have enough in the final third and lacked that bit of quality, but it is a point and if you cannot win these games, you certainly have to make sure you do not lose them.
"The (lack of goals) is a concern, and it has to be, but we also have to look at the positives.
"We have managed to stop a poor run, albeit not with wins."
Fulham boss Martin Jol on Dimitar Berbatov having to go off for treatment after being hit by Bradley Johnson's high boot:
"I get very irritated by that, when someone kicks you and you have to go off for five minutes.
"Later, Hugo Rodallega was also kicked and he had to go off, and they played again with 11 against 10.
"We have got to change that because it is an awful rule. You should send their player away as well for five minutes and play 10 against 10."
This page is best viewed in an up-to-date web browser with style sheets (CSS) enabled. While you will be able to view the content of this page in your current browser, you will not be able to get the full visual experience. Please consider upgrading your browser software or enabling style sheets (CSS) if you are able to do so.