The Scottish Labour leader has said there must be a Holyrood election following Humza Yousaf's resignation as First Minister, while the Scottish Greens called for "stability".
Scottish Labour have tabled a vote of no confidence in the Scottish Government as a whole, which is expected to take place this week.
Party leader Anas Sarwar said: "Scotland faces the biggest challenges since devolution but it now has a dysfunctional, chaotic and divided SNP Government.
"All this at a time when our country needs strong leadership to get us through the twin challenges of the economic crisis and the crisis in our NHS.
READ MORE: Tearful Yousaf announces his resignation as First Minister
READ MORE: Humza Yousaf resignation speech in full as SNP leader quits
"The SNP are a divided party which is out of ideas and incapable of rising to the challenges Scotland faces.
"They cannot impose another unelected First Minister on Scotland in a backroom deal, the people of Scotland should decide who leads our country.
"There must be an election - it's time for change and Scottish Labour is ready to deliver it."
He thanked Mr Yousaf for his public service and wished him and his family the best for the future.
Scottish Liberal Democrat leader Alex Cole-Hamilton echoed those sentiments and said Scotland "needs a new government".
He said: "Humza Yousaf's resignation hurls the SNP another step closer towards the end. This is a stale government that has been in power too long.
"Scotland needs a new government - one that won't make empty promises but will get the basics right."
He added: "We can't go on like this. This country can't have yet another First Minister without an election."
Mr Yousaf's former junior partners in government, the Scottish Greens, said now is the time to "return some stability".
The Scottish Greens co-leader, Patrick Harvie, said: "Humza Yousaf is right to resign. His position was no longer tenable after he broke the bonds of trust with the Scottish Greens and with everyone who wanted a stable, progressive, pro-independence government.
"It is regrettable that it has ended this way, it didn't need to. We draw no satisfaction or pleasure from this.
"But the Scottish Greens could no longer have confidence in Humza Yousaf after he chose to unilaterally end the Bute House Agreement.
"In doing so he let down the large majority of Scottish Green and SNP members who approved the agreement who wanted it to work.
"He chose to end a stable, majority government and jeopardised the progressive policy programme that both parties had committed to and were working to deliver.
"It is to his credit that he has taken personal responsibility. Now though is the time to return to some stability."
Scottish Conservative leader Douglas Ross, who tabled the motion of no confidence in Mr Yousaf as First Minister, said his party had "forced Humza Yousaf out of office for repeatedly failing Scotland".
He said: "Faced with our vote of no confidence, the SNP leader has quit rather than face a humiliating defeat."
He added: "Humza Yousaf is gone but the SNP remains - and the power to change that is in the hands of Scotland's voters.
"Now that we have forced Humza Yousaf out of office, we are asking voters to help us beat the SNP in seats up and down Scotland at the next general election."
Ash Regan, who leads the Alba Party in Holyrood, said it was "bizarre" that some SNP MSPs would rather he resigned from the top post than do a deal with the pro-independence party.
The former SNP politician added: "However, a new SNP leader and a new first minister will not change parliamentary arithmetic. I continue to stand ready to work in the best interests of Scotland and to advance the cause of Scottish independence."
Scottish Secretary Alister Jack said it was the "right thing" for Mr Yousaf to resign.
He said: "Humza Yousaf's leadership has lurched from crisis to crisis from the very start, and he could not command the confidence of the Scottish Parliament.
"Scotland now needs a stable, functioning Scottish Government focused on the issues that matter most to people: fixing public services and growing the economy."
The UK Government will work with Humza Yousaf's successor to deliver on "the real issues that matter to people", Downing Street has said.
Reacting to the First Minister's resignation, the Prime Minister's official spokesman said: "When the First Minister came to office he and the Prime Minister talked about wanting to work together to focus on the real issues that matter to people.
"I haven't, obviously, seen the news that you've reported on, but clearly if that's the case, the UK Government will work with the new administration to the same end, which is working together to deliver for people in Scotland, whether it's growing the economy, delivering jobs, enhancing energy security.
"For most people, they don't want to be distracted by the ins and outs of politics, they want to see their governments working together to deliver on their priorities."
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