Instances that I could find where batsman was not given out due to lack of appeal are as follows:
England skipper Andrew Strauss, who was the architect of India's agony by hitting his highest ODI score of 158, should have been out when he has scored a mere 13 if wicket-keeper Dhoni, Zaheer or close-in fielders had appealed for a caught behind at that point. Strauss clearly nicked a Zaheer delivery but there was no appeal by the Indians.
98 runs later, the England skipper had another stroke of luck when he again edged Zaheer to Dhoni with his score at 111 but none of the Indians on the field appealed.
94th over: England 367-7 (Moeen Ali 39, Broad 11)
Starc continues to Ali.
Ali appears to get a very slight edge to a ball and it was caught. No appeal, so not out!
View from Sportsmail's cricket correspondent Paul Newman in Cardiff:
And now it appears Moeen Ali got a very fine edge to the keeper in Starc's last over and nobody appealed! It's all happening out there! This is a brilliant start to the second day.
When he had 25, Amla pushed at a delivery from Stuart Broad. Of the England fielders, only Alastair Cook at first slip seemed convinced he had edged to the keeper. There was no appeal – only for Hot Spot to show a faint mark on Amla’s bat.
Desmond: ""EDIT: There was an edge! Oh dear..." Amla walks so that must have escaped everyone's attention " Batsmen walk when it suits them ... that's the general rule of thumb
Di: "What you really have to do is look at the replays, if Alma turned his head to the keeper, he knew he nicked it..." Maybe, maybe. Mind you, hard to moralise about non-walking batsmen in Eng v SA encounters at Trent Bridge, of all places...
Beaten past the outside edge. Or not. Hotspot picks up a little white blotch. Snicko detects a spike. Sarfraz dives across to his right and takes it. Faint edge, no appeal from Pakistan, they did not pick up the tickle.... They take a catch off Amir, but it all amounts to nothing
From espncricinfo article "Smith owns up to edge on 97": (Already stated in Nxaunxau answer)
"I was very surprised, it was pretty loud, obviously I was on 97, there was a fair bit going on in the crowd," Smith told ABC radio on Friday morning. "It was pretty loud, but yeah, I did nick it, and no one went up so I wasn't going to walk."
Attempting to avoid a short ball from Stephen Harmison, Waugh tried to withdraw his bat but not before the ball brushed it on the way to James Foster, England's wicketkeeper.
Foster half-heartedly held up his hands, as if not quite believing what he had seen, but along with the rest of the side did not ask the crucial question of umpire Dave Orchard.
From article "England simply not appealing in the field":
England captain Nasser Hussain: "I saw a deflection from mid-off but we couldn't hear a noise. Obviously James Foster started to appeal because he'd seen something and no one else heard anything. It came up on the big screen, looked like he'd gloved it. I was spewing, to be honest, [thinking] 'Why didn't we appeal?"'
Watch this youtube video. Sorry for the title.
11.1 Shorter in length on middle and leg, Vijay raises his bat and lets it go after looking to play at it first. Wade collects the ball and appeals. He is the only one doing so though. Replays though show there was a slight spike on Snicko. Oh dear, what a miss that for Australia. Was too faint a edge and hence nobody appealed along with Wade.
Oh hello. This has come off the glove. They don't appeal. Only half an appeal. Cummins has fallen in his follow-through so he doesn't know what happened, and Wade puts in only half a shout. Australia let another opportunity go
Check this video: Umpire Chris Gaffaney looked set to give India's Cheteshwar Pujara out on day four in Ranchi, but a half-hearted appeal may have changed his mind