This is not borne out by the record. Germany, for example, has had an equivalent system in place for decades, and has had the most stable governments of any democratic country during that time. The many other European countries with MMP have experienced no such instability either. Canada is one of the last holdouts in the world for Single Member Plurality (SMP), commonly known as First-Past-The Post.
We should look at the nature of a coalition, in any case. Many of the existing parties are coalitions in all but name; caucus bartering goes on all the time, attempting to reconcile varying interests across the country. Backroom dealing, which representatives of some of these very parties now profess to be afraid of, is an everyday fact of life in those parties, both inside and outside the legislature.
The necessity of forming formal coalitions will require a change in political culture, which is not a bad thing--more openness to compromise, more emphasis on win-win solutions. This is an attractive proposition on all sides of the political spectrum, but is frowned upon by those who simply want to impose their will on the province or the country with minority support.
In terms of public policy, MMP produces far more stability than does the present system. Under SMP, even a modest shift in the popular vote can result in massive policy changes, as one party or another acquires the necessary plurality to form a false majority (a majority of seats won with a minority of the popular vote). Under MMP, a modest shift in the popular vote simply produces an equally modest shift in the seat apportionment in the legislature.
2) MMP will allow small fringe parties to call the shots.
False. A threshold of 3% of the popular vote will keep out many of the single-issue extremists. And even if some achieve that threshold, it is far more likely that a major party will want to make coalitions with other large parties where compromise is both possible and more agreeable. There is no history of fringe parties holding the balance of power in European countries with MMP.
3) MMP will elect members who represent no one and whom no one's ever heard of.
False. Those members represent those who voted for their party. They are in the legislature because electors put them there. They are accountable to those electors as a whole.
Experience indicates that the vast majority of list members have also run in constituencies. In Germany, as Quebec scholar Louis Massicotte explains,
Typically, a list member starts out by running unsuccessfully in a constituency. To run, he or she has to become familiar with the local issues. The person tries again in the next election. If his or her party comes to power, its number of list seats will decline noticeably and the only way to get elected will likely be by running in a constituency. For this reason, such a person will remain active in the constituency during his or her term of office and give such activities almost as much effort as a “directly” elected member. . . the phenomenon is recognized in official literature for the public and some parliamentary websites even explicitly indicate the constituency in which each list member works. (Federal Parliament, provincial parliaments of North Rhine-Westphalia and Lower Saxony.)  For example, the 1998 federal election saw a major constituency shift. Victorious in 221 constituencies in 1994, the CDU/CSU won only 112 in 1998. Meanwhile, the SPD went from 103 to 212 direct seats. No fewer than 124 members changed category: 73 incumbent list members (all from the SPD, except 2) became constituency members, whereas 51 incumbent constituency members (all from the CDU/CSU) held their seats thanks to party lists. 
4) MMP is less efficient that Single Member Plurality (First-Past-The-Post)
This frequently-heard claim is really code for, "We worked the system, we got our majority, and we can do what we like until the next election." If strongman politics is preferred, then SMP/FPTP is the system for you. Governments can rule with only minority support, and impose their policies upon the unwilling majority until the next ballot, when all that is needed to keep doing what they're doing is to get another minority share of the vote.
The ruthless efficiency of a minority-supported dictatorship-between-elections is less preferable, however, than a system where nearly every elector's choice translates into seats. If these choices produce a variety of representatives, the democratic approach is to look for compromise and consensus. The latter is not make-the-trains-run-on-time efficient, but it works best in the long run, and is more conducive to citizen involvement in their governance.
5) MMP does not require parties to explain how their lists are put together.
This claim, made recently by Jason Cherniak and others, is simply wrong. The Ontario Citizens' Assembly specifically recommended that parties be required to make public their method of list creation by submitting their selection process to the non-partisan Elections Ontario, which would then publish that information widely. Electors could see, for example, if backroom party hacks or cronies of the Premier have been awarded the list positions, or if, on the other hand, the parties have a more democratic and open process, that does something, for example, about the representation of women and minorities, geographic balance, and so on.
6) MMP will make contact with your representative more difficult.
It is worth noting that at present the vast majority of citizens a) never contact their political representative, and b) vote for the party, not the candidate. But MMP will not make contacting one's MLA more difficult--indeed, the reverse is true.
For those with short memories, Ontario had 130 ridings until Mike Harris sliced that back to 103 in 1996 with the Fewer Politicians Act. The current MMP proposal would restore almost all of those seats (129). The number of ridings would be slightly reduced (to 90 from the current 107). The rest would be apportioned to create a legislature that actually reflects the way people voted.
Under MMP, citizens who want to contact their representative between elections now have a choice: to go the riding MLA, or, if that person is not to their political taste, to approach a list member from their party of choice.
7) MMP is confusing.
Wrong. What's confusing is the welter of lies, distortions and misapprehensions emanating from the special interests who oppose MMP. MMP is in fact simplicity itself: an elector gets two votes, one for the party of choice, and one for the riding candidate of choice. Once the riding contests are decided, the popular vote determines how list seats are handed out. End of story. Not rocket science, but common democratic sense.
8) MMP will produce two tiers of political representative.
Once again, this has not been the case in Germany, with its long history of MMP. As Eckhard Jesse, cited by Louis Massicotte, points out,
The assumption, that the two-vote system produces two kinds of MP, the constituency MP and the Landesliste MP, is empirically refutable. Contrary to widespread opinions, it is of absolutely no importance whether a mandate is obtained through the constituency and the Landesliste. Double candidatures are the rule. The voters do not perceive the difference at all. 
9) MMP is undemocratic.
"MMP gives parties too much power!" This is a bit rich, coming from those who represent parties that prefer to go on ruling with minority popular support, whose candidates can be shoehorned by the party leader into constituencies over the objections of the local riding associations, and all of whose riding candidates in any case are selected by the party.
So now the party lists will also be selected by the party--no change there. And the parties, as noted, will have to make the public aware of their process of choosing list candidates. Any Ignatieff or Anders shenanigans on a list-wide basis will almost certainly meet with elector resistance.
A system that produces majorities in Ontario that since the early 1930s have had only minority support is not democratic. A system that allows the party that gets fewer votes than their rival to form a majority government (as happened recently in New Brunswick) isn't democratic. A system that prevents political views held by a substantial number of electors from being represented in the legislature because they aren't concentrated in a handful of ridings is undemocratic. A system that produces wide swings in policy when only a minor shift in public opinion has taken place is undemocratic.
MMP, on the other hand, is a significant move in the direction of democratic governance. An elector's vote will make a difference: even in a solid Liberal riding, a vote for the Greens or the NDP will count. The legislature, as noted, will come much closer to reflecting the range and the relative strengths of the political views held by the electorate. So-called "strategic voting," in which electors are tempted to vote for a party they don't want to keep out another party they want even less, will become unnecessary, and they can vote for the party they do want, knowing that their votes will be counted. Now, that's democratic.
10) MMP is divisive.
This claim is based upon the false notion that a large number of parties will create chaos in the legislature--something not borne out by actual experience. But let's look at the divisiveness created and fostered by the present system.
First-Past-The-Post encourages a narrow regionalism rather than national or provincial consensus. In order to have a chance against an established, entrenched party like the Liberals, another party will lean towards a strategy of concentrating its riding votes by exaggerating regional differences. We have seen this clearly on the federal stage in the case of both the Reform Party and the Bloc Québécois. In the former case, a coalition came into being, but the Conservative Party of Canada still trades on Western alienation. In Ontario, we have northern vs. southern and urban vs. rural; even in Toronto we have the "905" folks vs. the "416" people.
MMP, on the other hand, will yield roughly proportional results that make riding concentration unnecessary. The efforts of political parties could be re-directed towards the actual issues and towards consensus instead of division. None of this is to say, of course, that genuine regional differences and interests will disappear--only that they will be less likely to be opportunistically used and abused by parties vying for seats.
MMP is not a panacea, and will not produce a democratic utopia. There are far too many other aspects of our system of governance that would have to be looked at--for example, why should the Premier be chosen by the party instead of by the electorate? Why should the Cabinet be appointed by the Premier instead of by the legislature as a whole? How can Aboriginal interests be effectively represented in the legislature under either the current system or MMP? These are other questions, for other debates, but in the meantime MMP is clearly a step in the right direction. To let this opportunity slip through our fingers because of the deliberate prevarication of entrenched special interests would be a tragedy. Let's not let that happen.