According to initial unofficial results, Fatah did well in many villages in the West Bank, wining as many as 50 of the 84 contested councils.
However, Hamas emerged the winner in large towns such as Rafah, Qalqilya, Bethlehem, Buraij and Saair near Hebron, and seems to have won as many as 32 local councils in the West Bank.
Hamas leaders have argued, justifiably, that while Fatah won significantly more council seats than Hamas, the Islamist movement actually surpassed Fatah in terms of the number of votes it received.
Hamas spokesman in Gaza Sami Abu Zuhri alluded to this during a news conference on Saturday night.
He said: "A seat in a small village or hamlet of a few hundred voters is not equal to a seat in a large town such as Rafah with tens of thousands of voters."
Another serious problem marring the elections has been the status and political affiliation of independent candidates.
Fatah claimed most of these candidates were affiliated with the movement, although many of them were either clan or truly independent candidates.
Similarly, Hamas has claimed that many of the religious-oriented independents were members of Hamas and that they did not declare their Hamas identity for security reasons, an allusion to possible arrest by the Israeli occupation authorities.
Hamas proved popular in built up
areas rather than villages
The confusion over the independents has prompted the Palestinian election steering committee to postpone the publication of the results until Monday.
Moreover, according to committee officials, only the names of winning candidates will be announced, irrespective of their political affiliations.
"We will announce the names of the winners, and then Hamas and Fatah can quarrel among themselves over the political affiliation of the winners," said Saad Muhammad, of the steering committee.
Earlier, Hamas accused the committee of being "at Fatah's beck and call" when some committee officials announced that Fatah's lists won up to 60% of the contested councils when actually many of the winning candidates in many localities were independents.
Some political analysts in the West Bank have described the election as inconclusive.
"The balance of power between Fatah and Hamas has not changed. It is true that Fatah has recovered some of its lost esteem and stature, but we by no means can say that Hamas has been defeated," said Hani al-Masri, a columnist and political analyst.
Al-Masri said the outcome showed Hamas could still pose a serious challenge to Fatah in the upcoming legislative elections.
"The balance of power between Fatah and Hamas has not changed"
Hani al-Masri, columnist
and political analyst
He noted that Hamas could still "surprise us" in the big towns such as Nablus, Hebron, Ram Allah, Gaza, Jenin and Tulkaram.
The PA has so far refused to allow elections to take place in these towns, possibly for fear Hamas might win, which could then negatively affect Fatah for the parliamentary election this summer.
Some Palestinian intellectuals hope the good performance by Fatah in last week's election will encourage the movement to agree to hold the legislative elections on time, on 17 July.
However, al-Masri thinks Fatah may be dragging its feet to postpone the elections, not for procedural reasons, but for political ones.
Some Palestinian sources close to the PA leadership have intimated to Aljazeera.net that US Secretary of State Condoleezza Rice has privately asked PA President Mahmoud Abbas to "find a way to forestall a possible victory by Hamas in the upcoming elections".
According to the sources, Abbas refused the idea, arguing that elections ought to be held on time regardless of who would win.
Last week, Hamas leaders in the West Bank revealed Abbas had dispatched "a messenger" proposing a "limited postponement" of the elections in exchange for the creation of a national unity government.
Hamas leaders rejected the idea, saying the elections should be held on time.